Bayesian Analysis Using Joint Progressive Type-II Censoring Scheme

Abstract

The joint censoring technique becomes crucial when the study’s aim is to assess the comparative advantages of products concerning their service times. In recent years, there has been a growing interest in progressive censoring as a means to reduce both cost and experiment duration. This article delves into the realm of statistical inference for the three-parameter Burr-XII distribution using a joint progressive Type II censoring approach applied to two separate samples. We explore both maximum likelihood and Bayesian methods for estimating model parameters. Furthermore, we derive approximate confidence intervals based on the observed information matrix and employ four bootstrap methods to obtain confidence intervals. Bayesian estimators are presented for both symmetric and asymmetric loss functions. Since closed-form solutions for Bayesian estimators are unattainable, we resort to the Markov chain Monte Carlo method to compute these estimators and the corresponding credible intervals. To assess the performance of our estimators, we conduct extensive simulation experiments. Finally, to provide a practical illustration, we analyze a real dataset.

1. Introduction

The Burr-XII distribution, originally introduced by Burr [1], has found extensive applications in various domains, including lifetime modeling for reliability analysis, addressing life-testing challenges, and devising acceptance sampling plans, as exemplified in the works of Abbasi et al. [2] and other researchers. It has also been effectively utilized in the analysis of observational data across diverse fields such as meteorology, finance, and hydrology, as showcased in studies conducted by Chen et al. [3], Ali and Jaheen [4], Burr [1] and Lio et al. [5]. Moreover, Shao et al. [6] delved into the modeling of extreme events using the three-parameter Burr-XII distribution (TPBXIID), notably in the context of flood-frequency analysis.

Our decision to employ the TPBXIID stems from its remarkable adaptability, spanning a wide spectrum of shapes, from highly skewed to nearly symmetric. This versatility renders it a valuable model for datasets that do not adhere to standard shapes. Notably, the distribution’s three parameters, denoted as $\alpha$, $\gamma$, and $\theta$, offer straightforward interpretations, simplifying the analysis of statistical outcomes and enabling comparisons across different datasets. As a distribution limited to non-negative values, the Burr-XII distribution is frequently harnessed for modeling data related to lifetimes, sizes, or quantities. It consistently demonstrates an excellent fit to empirical datasets and, in specific data types, is known to outperform other commonly employed distributions, including the Weibull distribution.

Furthermore, the Burr-XII distribution serves as a generalized form encompassing several other distributions, including the Lomax (Pareto II), Burr-XII, and log-logistic distributions. In summary, the three-parameter Burr-XII distribution emerges as a versatile tool for modeling non-negative data, which are characterized by a broad spectrum of shapes. Its flexibility and interpretability render it a popular choice in statistical modeling.

Cook and Johnson [7] applied the Burr model to attain superior fits for a uranium survey dataset, while Zimmer et al. [8] delved into the statistical and probabilistic properties of the Burr-XII distribution and its relationships with other distributions commonly employed in reliability analyses. Additionally, Tadikamalla [9] extended the two-parameter Burr-XII distribution by introducing an additional scale parameter, resulting in the TPBXIID. This extension has sparked increased interest in the applications of the Burr-XII distribution.

Tadikamalla also established mathematical connections among Burr-related distributions, revealing that the Lomax distribution constitutes a special instance of the Burr-XII distribution, and the compound Weibull distribution represents a generalization of the Burr distribution. Furthermore, he demonstrated that several widely used distributions, including Weibull, logistic, log-logistic, normal, and lognormal distributions, can be viewed as specific cases of the Burr-XII distribution by appropriately configuring the distribution parameters.

In essence, the TPBXIID proves to be highly adaptable, encompassing two shape parameters and one scale parameter in the distribution function, thereby allowing it to represent a diverse range of distribution shapes. The TPBXIID can be characterized through its cumulative distribution function (CDF) and probability density function (PDF), as expressed, respectively, in Equations (1) and (2).

$$F(x;\alpha ,\theta ,\gamma )=1-{\left[1+{\left(\frac{x}{\alpha }\right)}^{\theta }\right]}^{-\gamma };x>0,\alpha ,\theta ,\gamma >0$$

(1)

$$f(x;\alpha ,\theta ,\gamma )=\theta \gamma {\alpha }^{-\theta }{x}^{\theta -1}{\left[1+{\left(\frac{x}{\alpha }\right)}^{\theta }\right]}^{-(\gamma +1)};x>0,\alpha ,\theta ,\gamma >0.$$

(2)

where $\gamma$ and $\theta$ represent the shape parameters, and $\alpha$ serves as the scale parameter. Notably, when $\theta >1$, the density function exhibits an upside-down bathtub shape (unimodal) with the mode located at $x=\alpha {\left[(\theta -1)/(\theta \gamma +1)\right]}^{\frac{1}{\theta }}$, while it assumes an L-shaped form when $\theta =1$.

Recently, Mead and Afify [10] ventured into defining and examining the properties and applications of the five-parameter Burr-XII distribution, which is referred to as the Kumaraswamy exponentiated Burr-XII. Moreover, Shafqat et al. [11] explored the utilization of moving average control charts in the context of Burr X and inverse Gaussian, and Aslam et al. [12] studied a new generalized Burr-XII distribution with real-life applications.

The joint censoring approach proves to be a valuable and practical method for comparing life tests of products originating from various units within the same facility. Consider a scenario where two production lines operate within the same facility, generating products. In this setup, two independent samples of sizes m and n can be selected from each production line and subjected to simultaneous life-testing experiments. To optimize resource utilization, reduce costs, and save time, researchers often employ a joint progressive Type-II censoring scheme (JP-II-CS). This approach is instrumental in terminating life testing when a predetermined number of failures are observed.

Numerous studies in the literature have explored JP-II-CS and inference methods associated with it. For example, Rasouli and Balakrishnan [13] introduced likelihood inference techniques for two exponential distributions based on JP-II-CS. Doostparast et al. [14] delved into Bayes estimation under the linear exponential loss function using JP-II-CS data. Balakrishnan et al. [15] provided likelihood inference procedures for k exponential distributions under JP-II-CS, while Mondal and Kundu [16] focused on the point and interval estimation of Weibull parameters within the context of JP-II-CS.

Goel and Krishna [17] explored likelihood and Bayesian inference for k Lindley populations under a joint Type-II censoring scheme. Krishna and Goel [18] conducted a study on Lindley populations utilizing JP-II-CS. Additionally, Goel and Krishna [19] discussed statistical inference for two Lindley populations under a balanced JP-II-CS. Bayoud and Raqab [20] investigated classical and Bayesian inferences for two Topp–Leone models under JP-II-CS, while Chen and Gui [21] addressed the statistical inference of the generalized inverted exponential distribution in the context of JP-II-CS.

Pandey and Srivastava [22] focused on Bayesian inference for two log-logistic populations under JP-II-CS, and Qiao and Gui [23] tackled the statistical inference of the Weighted Exponential Distribution under similar censoring conditions.

Recently, Hassan et al. [24] delved into the statistical inference of the Burr Type III distribution under joint progressively Type II censoring. Kumar and Kumari [25] explored Bayesian and likelihood estimation techniques for two inverse Pareto populations under joint progressive censoring conditions.

According to Rasouli and Balakrishnan [13], JP-II-CS is described as follows. Let $X_1\dots ,X_m$, be lifetimes of m units for product A, and they are supposed to be independent and identically distributed (iid) random variables from TPBXIID with a CDF given by

$$F_1(x;{\alpha }_1,{\theta }_1,{\gamma }_1)=1-{\left[1+{\left(\frac{x}{{\alpha }_1}\right)}^{{\theta }_1}\right]}^{-{\gamma }_1},x>0,{\alpha }_1,{\theta }_1,{\gamma }_1>0,$$

(3)

and the PDF is

$$f_1(x;{\alpha }_1,{\theta }_1,{\gamma }_1)={\theta }_1{\gamma }_1{\alpha }_1^{-{\theta }_1}{x}^{{\theta }_1-1}{\left[1+{\left(\frac{x}{{\alpha }_1}\right)}^{{\theta }_1}\right]}^{-{\gamma }_1-1},x>0,{\alpha }_1,{\theta }_1,{\gamma }_1>0.$$

(4)

In a similar manner, consider a set of n lifetimes denoted as $Y_1,Y_2,\dots ,Y_n$ for the product B. These lifetimes correspond to n units and are treated as iid random variables following the TPBXIID. The CDF for this distribution is given by:

$$F_2(y;{\alpha }_2,{\theta }_2,{\gamma }_2)=1-{\left[1+{\left(\frac{y}{{\alpha }_2}\right)}^{{\theta }_2}\right]}^{-{\gamma }_2},y>0,{\alpha }_2,{\theta }_2,{\gamma }_2>0,$$

(5)

and PDF is

$$f_2(y;{\alpha }_2,{\theta }_2,{\gamma }_2)={\theta }_2{\gamma }_2{\alpha }_2^{-{\theta }_2}{y}^{{\theta }_2-1}{\left[1+{\left(\frac{y}{{\alpha }_2}\right)}^{{\theta }_2}\right]}^{-{\gamma }_2-1},y>0,{\alpha }_2,{\theta }_2,{\gamma }_2>0.$$

(6)

Where ${\theta }_1$, ${\gamma }_1$, ${\theta }_2$ and ${\gamma }_2$ are shape parameters and ${\alpha }_1$ and ${\alpha }_2$ are scale parameters. In this scenario, let $K=m+n$ denote the total sample size and ${\lambda }_1\le \dots \le {\lambda }_K$ indicate the order statistics of the K random variables $\{X_1,X_2,\dots ,X_m;Y_1,Y_2,\dots ,Y_n\}$. The JP-II-CS method is applied as follows: when the first failure occurs, $R_1$ units are randomly removed from the remaining $K-1$ surviving units. The same process is repeated for the second failure, where $R_2$ units are randomly withdrawn from the remaining $K-R_1-2$ surviving units, and so on. At the $rth$ failure, all remaining $R_r=K-r-{\sum }_{i=1}^{r-1}{R}_i$ surviving units are withdrawn from the experiment. The JP-II-CS is represented by $R=(R_1,R_2,\dots ,R_r)$, and the total number of failures r is predetermined before conducting the experiment. Suppose that $R_i=S_i+T_i$, $i=1,\dots ,r,$ where $S_i$ and $T_i$ represent the number of units withdrawn at the time of $ith$ failure related to X and Y samples, respectively. These values are unknown and random variables. The data observed in this form will consist of $(H,\lambda ,S)$, where $H=(H_1,H_2,\dots ,H_r)$, where $H_i=1$ or 0 if ${\lambda }_i$ comes from X or Y failure, respectively, $\lambda =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_r)$ with $r<K$ and $S=(S_1,S_2,\dots ,S_r)$.

In this study, we employ a JP-II-CS strategy to formulate statistical inferences and assess two independent samples from the TPBXIID. We derive point and interval estimators using Bayesian and maximum likelihood estimation (MLE) techniques. Subsequently, we calculate asymptotic confidence intervals (ACI) based on the observed information matrix. These confidence intervals (CIs) are computed through various bootstrap techniques, including Bootstrap-P (Boot-P), Bootstrap-T (Boot-T), Bias-Corrected Bootstrap (Boot-BC), and Bias-Corrected Accelerated Bootstrap (Boot-BCa) methods. We assume a gamma prior distribution for both the shape and scale parameters. Employing the Metropolis–Hastings (M-H) method, we obtain Bayes estimates and credible intervals (CRIs) for the informative prior under both the squared error (SE) and linear exponential (LINEX) loss functions. To evaluate the effectiveness of these diverse approaches, we conduct Monte Carlo simulations and analyze real-world data.

The paper is structured in the following way: Section 2 outlines the derivation of the MLEs for the unknown parameters of TPBXIID. In Section 3, there is a presentation of ACIs that depend on the MLEs. Section 4 discusses different bootstrap CIs. The Bayesian analysis is performed in Section 5. To illustrate the estimation methods developed in this paper, we analyze real datasets in Section 6. The results of simulation are presented in Section 7. Finally, a brief conclusion can be found in Section 8.

2. Classical Likelihood Estimation

Based on the work of Rasouli and Balakrishnan [13], the likelihood function for $(H,\lambda ,S)$ can be expressed as follows:

$$L(.)=C\prod _{i=1}^r\left[{\left[f\left({\lambda }_i\right)\right]}^{h_i}{\left[q\left({\lambda }_i\right)\right]}^{1-h_i}\right]{\left[\overline{F}\left({\lambda }_i\right)\right]}^{s_i}{\left[\overline{Q}\left({\lambda }_i\right)\right]}^{t_i},$$

(7)

where $L(.)L({\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,{\gamma }_2;H,\lambda ,S)$, $w_1\le w_2\le \dots \le w_r$,   $\overline{F}=1-F$,   $\overline{Q}=1-Q$,   ${\sum }_{i=1}^r{s}_i=m-m_r$,   ${\sum }_{i=1}^r{t}_i=n-n_r$,   ${\sum }_{i=1}^r{s}_i+{\sum }_{i=1}^r{t}_i={\sum }_{i=1}^r{R}_i$ and $C=D_1{D}_2$ with

$$D_1=\prod _{j=1}^r\left\{\left(m-\sum _{i=1}^{j-1}{h}_i-\sum _{i=1}^{j-1}{s}_i\right)z_j+\left(n-\sum _{i=1}^{j-1}(1-h_i)-\sum _{i=1}^{j-1}(R_i-s_i)\right)(1-z_j)\right\},$$

$$D_2=\prod _{j=1}^r\left\{\frac{\left(\begin{array}{c}m-{\sum }_{i=1}^{j-1}{h}_i-{\sum }_{i=1}^{j-1}{s}_i\\ s_i\end{array}\right)\left(\begin{array}{c}n-{\sum }_{i=1}^{j-1}(1-h_i)-{\sum }_{i=1}^{j-1}(R_i-s_i)\\ t_i\end{array}\right)}{\left(\begin{array}{c}m+n-j-{\sum }_{i=1}^{j-1}{R}_i\\ R_i\end{array}\right)}\right\}.$$

The likelihood function in (7) becomes

$$\begin{array}{cc}\hfill L(.)=& C{\left(\frac{{\theta }_1{\gamma }_1}{{\alpha }_1}\right)}^{m_r}{\left(\frac{{\theta }_2{\gamma }_2}{{\alpha }_2}\right)}^{n_r}{\alpha }_1^{-m_r({\theta }_1-1)}{\alpha }_2^{-n_r({\theta }_2-1)}\hfill \\ & \times \prod _{i=1}^r[{\lambda }_i^{({\theta }_1-1)h_i}{\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_1}\right)}^{{\theta }_1}\right]}^{-({\gamma }_1+1)h_i}{\lambda }_i^{({\theta }_2-1)(1-h_i)}\hfill \\ \hfill \times & {\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_2}\right)}^{{\theta }_2}\right]}^{-({\gamma }_2+1)(1-h_i)}]{\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_1}\right)}^{{\theta }_1}\right]}^{-{\gamma }_1{s}_i}{\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_2}\right)}^{{\theta }_2}\right]}^{-{\gamma }_2{t}_i}.\hfill \end{array}$$

(8)

The log-likelihood function for the TPBXIID, as related to Equation (8), is as follows:

$$\begin{array}{cc}\hfill \ell (.)=& lnC+m_{r}ln{\theta }_1+m_{r}ln{\gamma }_1-m_{r}ln{\alpha }_1+n_{r}ln{\theta }_2+n_{r}ln{\gamma }_2-n_{r}ln{\alpha }_2+\hfill \\ & ({\theta }_1-1)\left[\sum _{i=1}^r{h}_{i}ln{\lambda }_i-m_{r}ln{\alpha }_1\right]+({\theta }_2-1)\left[\sum _{i=1}^r(1-h_i)ln{\lambda }_i-n_{r}ln{\alpha }_2\right]\hfill \\ & -\sum _{i=1}^r\left\{\left({\gamma }_1(h_i+s_i)+h_i\right)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_1}\right)}^{{\theta }_1}\right]\right\}\hfill \\ & -\sum _{i=1}^r\left\{\left({\gamma }_2(t_i-h_i+1)-h_i+1\right)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_2}\right)}^{{\theta }_2}\right]\right\}.\hfill \end{array}$$

(9)

Deriving the first derivative of Equation (9) concerning ${\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,\mathrm{and},{\gamma }_2$ and setting each one as follows:

$$\frac{-m_r{\widehat{\theta }}_1}{{\widehat{\alpha }}_1}+\sum _{i=1}^r\frac{\left({\gamma }_1(h_i+s_i)+h_i\right){\lambda }_i{\widehat{\theta }}_1{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1-1}}{{\widehat{\alpha }}_1^2\left[1+{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1}\right]}=0,$$

(10)

$$\frac{-n_r{\widehat{\theta }}_2}{{\widehat{\alpha }}_2}+\sum _{i=1}^r\frac{\left({\gamma }_2(t_i-h_i+1)-h_i+1\right){\lambda }_i{\widehat{\theta }}_2{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2-1}}{{\widehat{\alpha }}_2^2\left[1+{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2}\right]}=0,$$

(11)

$$\frac{m_r}{{\widehat{\theta }}_1}-m_{r}ln{\widehat{\alpha }}_1+\sum _{i=1}^r{h}_{i}ln{\lambda }_i-m_{r}ln{\alpha }_1-\sum _{i=1}^r\frac{\left[{\widehat{\gamma }}_1(h_i+s_i)+h_i\right]{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1}ln\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}{1+{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1}}=0,$$

(12)

$$\frac{n_r}{{\widehat{\theta }}_2}-n_{r}ln{\widehat{\alpha }}_2+\sum _{i=1}^r(1-h_i)ln{\lambda }_i-n_{r}ln{\alpha }_2-\sum _{i=1}^r\frac{\left[{\widehat{\gamma }}_2(t_i-h_i+1)-h_i+1\right]{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2}ln\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}{1+{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2}}=0,$$

(13)

$$\frac{m_r}{{\widehat{\gamma }}_1}-\sum _{i=1}^r(h_i+s_i)ln\left[1+{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1}\right]=0,$$

(14)

$$\frac{n_r}{{\widehat{\gamma }}_2}-\sum _{i=1}^r(t_i-h_i+1)ln\left[1+{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2}\right]=0.$$

(15)

From (14) and (15), we obtain the MLE $\widehat{\gamma }$ as

$${\widehat{\gamma }}_1=m_r{\left[\sum _{i=1}^r(h_i+s_i)ln[1+{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1}]\right]}^{-1},$$

(16)

$${\widehat{\gamma }}_2=n_r{\left[\sum _{i=1}^r(t_i-h_i+1)ln[1+{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2}]\right]}^{-1}.$$

(17)

To acquire the MLEs for ${\alpha }_1,{\alpha }_2,{\theta }_1,\mathrm{and},{\theta }_2$, the resolution of Equations (10)–(13) is necessary. Nevertheless, attaining analytical solutions for these equations proves highly demanding, making it arduous to obtain closed-form expressions for each parameter. Consequently, resorting to a numerical approach, such as the Newton–Raphson iteration method, becomes imperative to estimate the parameter values.

3. Approximate Confidence Interval

The second derivative of log-likelihood function with respect to ${\alpha }_1,{\alpha }_2,{\theta }_1,\mathrm{and},{\theta }_2$, gives

$$\begin{array}{cc}\hfill & \frac{{\partial }^2\ell }{\partial {\alpha }_1^2}=\frac{m_r{\widehat{\theta }}_1}{{\widehat{\alpha }}_1^2}-\sum _{i=1}^r\frac{\left({\gamma }_1(h_i+s_i)+h_i\right){\lambda }_i{\widehat{\theta }}_1{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1-1}\left[{\widehat{\alpha }}_1^2(1-{\widehat{\theta }}_1){\mathrm{\Omega }}_1+2{\widehat{\alpha }}_1{\mathrm{\Omega }}_1+{\widehat{\theta }}_1{\lambda }_i\right]}{{\left({\widehat{\alpha }}_1^2{\mathrm{\Omega }}_1\right)}^2},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & \frac{{\partial }^2\ell }{\partial {\alpha }_2^2}=\frac{n_r{\widehat{\theta }}_2}{{\widehat{\alpha }}_2^2}-\sum _{i=1}^r[\frac{\left({\gamma }_2(t_i-h_i+1)-h_i+1\right){\lambda }_i{\widehat{\theta }}_2{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2-1}\left[{\widehat{\alpha }}_2^2(1-{\widehat{\theta }}_2){\mathrm{\Omega }}_2+2{\widehat{\alpha }}_2{\mathrm{\Omega }}_2+{\widehat{\theta }}_2{\lambda }_i\right]}{{\left({\widehat{\alpha }}_2^2{\mathrm{\Omega }}_2\right)}^2},\hfill \end{array}$$

(19)

$$\frac{{\partial }^2\ell }{\partial {\theta }_1^2}=\frac{-m_r}{{\widehat{\theta }}_1^2}-\sum _{i=1}^r\frac{\left[{\widehat{\gamma }}_1(h_i+s_i)+h_i\right]{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1}{\left[ln\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)\right]}^2}{{\mathrm{\Omega }}_1^2},$$

(20)

$$\frac{{\partial }^2\ell }{\partial {\theta }_2^2}=\frac{-n_r}{{\widehat{\theta }}_2^2}-\sum _{i=1}^r\frac{\left[{\widehat{\gamma }}_2(t_i-h_i+1)-h_i+1\right]{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2}{\left[ln\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)\right]}^2}{{\mathrm{\Omega }}_2^2},$$

(21)

$$\begin{array}{cc}\hfill & \frac{{\partial }^2\ell }{\partial {\theta }_1\partial {\alpha }_1}=\frac{{\partial }^2\ell }{\partial {\alpha }_1\partial {\theta }_1}=\frac{-m_r}{{\widehat{\alpha }}_1}+\sum _{i=1}^r[\frac{\left({\widehat{\gamma }}_1(h_i+s_i)+h_i\right){\widehat{\theta }}_1{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1}ln\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right){\mathrm{\Omega }}_1}{{\mathrm{\Omega }}_1^2}\hfill \\ & +\frac{\left({\widehat{\gamma }}_1(h_i+s_i)+h_i\right)\left[\frac{1}{{\widehat{\alpha }}_1}{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1}{\mathrm{\Omega }}_1+\left(\frac{{\widehat{\theta }}_1}{{\widehat{\alpha }}_1}\right){\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{2{\widehat{\theta }}_1}ln\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)\right]}{{\mathrm{\Omega }}_1^2}],\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & \frac{{\partial }^2\ell }{\partial {\theta }_2\partial {\alpha }_2}=\frac{{\partial }^2\ell }{\partial {\alpha }_2\partial {\theta }_2}=\frac{-n_r}{{\widehat{\alpha }}_2}+\sum _{i=1}^r[\frac{\left({\widehat{\gamma }}_2(t_i-h_i+1)-h_i+1\right){\widehat{\theta }}_2{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2}ln\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right){\mathrm{\Omega }}_2}{{\mathrm{\Omega }}_2^2}\hfill \\ & +\frac{\left({\gamma }_2(t_i-h_i+1)-h_i+1\right)\left[\frac{1}{{\widehat{\alpha }}_2}{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2}{\mathrm{\Omega }}_2+\left(\frac{{\widehat{\theta }}_2}{{\widehat{\alpha }}_2}\right){\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{2{\widehat{\theta }}_2}ln\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)\right]}{{\mathrm{\Omega }}_2^2}],\hfill \end{array}$$

(23)

$$\frac{{\partial }^2\ell }{\partial {\gamma }_1\partial {\alpha }_1}=\frac{{\partial }^2\ell }{\partial {\alpha }_1\partial {\gamma }_1}=\sum _{i=1}^r\frac{\left[(h_i+s_i)\right]{\lambda }_i{\widehat{\theta }}_1{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1-1}}{{\widehat{\alpha }}_1^2{\mathrm{\Omega }}_1},$$

(24)

$$\frac{{\partial }^2\ell }{\partial {\gamma }_1\partial {\theta }_1}=\frac{{\partial }^2\ell }{\partial {\theta }_1\partial {\gamma }_1}=-\sum _{i=1}^r\frac{\left[(h_i+s_i)\right]{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1}ln\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}{{\mathrm{\Omega }}_1},$$

(25)

$$\frac{{\partial }^2\ell }{\partial {\gamma }_2\partial {\alpha }_2}=\frac{{\partial }^2\ell }{\partial {\alpha }_2\partial {\gamma }_2}=\sum _{i=1}^r\frac{\left[(t_i-h_i+1)\right]{\lambda }_i{\widehat{\theta }}_2{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2-1}}{{\widehat{\alpha }}_2^2{\mathrm{\Omega }}_2},$$

(26)

$$\frac{{\partial }^2\ell }{\partial {\gamma }_2\partial {\theta }_2}=\frac{{\partial }^2\ell }{\partial {\theta }_2\partial {\gamma }_2}=-\sum _{i=1}^r\frac{\left[(t_i-h_i+1)\right]{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2}ln\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}{{\mathrm{\Omega }}_2},$$

(27)

$$\frac{{\partial }^2\ell }{\partial {\gamma }_1^2}=\frac{-m_r}{{\widehat{\gamma }}_1^2},$$

(28)

$$\frac{{\partial }^2\ell }{\partial {\gamma }_2^2}=\frac{-n_r}{{\widehat{\gamma }}_2^2},$$

(29)

and

$$\begin{array}{cc}\hfill & \frac{{\partial }^2\ell }{\partial {\alpha }_1\partial {\alpha }_2}=\frac{{\partial }^2\ell }{\partial {\alpha }_2\partial {\alpha }_1}=\frac{{\partial }^2\ell }{\partial {\alpha }_1\partial {\theta }_2}=\frac{{\partial }^2\ell }{\partial {\theta }_2\partial {\alpha }_1}=\frac{{\partial }^2\ell }{\partial {\alpha }_1\partial {\gamma }_2}=\frac{{\partial }^2\ell }{\partial {\gamma }_2\partial {\alpha }_1}=\frac{{\partial }^2\ell }{\partial {\alpha }_2\partial {\theta }_1}=\frac{{\partial }^2\ell }{\partial {\theta }_1\partial {\alpha }_2}=\frac{{\partial }^2\ell }{\partial {\alpha }_2\partial {\gamma }_1}=\hfill \\ & \frac{{\partial }^2\ell }{\partial {\gamma }_1\partial {\alpha }_2}=\frac{{\partial }^2\ell }{\partial {\theta }_1\partial {\theta }_2}=\frac{{\partial }^2\ell }{\partial {\theta }_2\partial {\theta }_1}=\frac{{\partial }^2\ell }{\partial {\theta }_1\partial {\gamma }_2}=\frac{{\partial }^2\ell }{\partial {\gamma }_2\partial {\theta }_1}=\frac{{\partial }^2\ell }{\partial {\theta }_2\partial {\gamma }_1}=\frac{{\partial }^2\ell }{\partial {\gamma }_1\partial {\theta }_2}=\frac{{\partial }^2\ell }{\partial {\gamma }_1\partial {\gamma }_2}=\frac{{\partial }^2\ell }{\partial {\gamma }_2\partial {\gamma }_1}=0,\hfill \end{array}$$

(30)

where

$${\mathrm{\Omega }}_1=1+{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_1}\right)}^{{\widehat{\theta }}_1}\mathrm{and}{\mathrm{\Omega }}_2=1+{\left(\frac{{\lambda }_i}{{\widehat{\alpha }}_2}\right)}^{{\widehat{\theta }}_2}.$$

The asymptotic variances–covariances of the MLEs for parameters ${\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1, \mathrm{and} {\gamma }_2$ are given by elements of the inverse of the Fisher information matrix (FIM) where the FIM is obtained by taking the expectation of minus Equations (18)–(29) and defined as

$$I_{ij}=-E\left(\begin{array}{c}\frac{{\partial }^2\ell }{\partial {\delta }_i\partial {\delta }_j}\end{array}\right),$$

(31)

where $i,j=1,2,3,4,5,6 \mathrm{and} ({\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4,{\delta }_5,{\delta }_6)=({\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1\mathrm{and}{\gamma }_2)$.

The mathematical expressions for the expectations referred to in Equation (31) lack precision and can be located in Cohen’s work [26], where he computed the asymptotic variance–covariance matrices for various sample types.

Therefore, we rely on the estimated asymptotic variance–covariance matrix for the MLEs.

$$\begin{array}{c}\hfill I({\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,{\gamma }_2)=\left(\begin{array}{cccccc}-\frac{{\partial }^2\ell }{\partial {\alpha }_1^2}& -\frac{{\partial }^2\ell }{\partial {\alpha }_1\partial {\alpha }_2}& -\frac{{\partial }^2\ell }{\partial {\alpha }_1\partial {\theta }_1}& -\frac{{\partial }^2\ell }{\partial {\alpha }_1\partial {\theta }_2}& -\frac{{\partial }^2\ell }{\partial {\alpha }_1\partial {\gamma }_1}& -\frac{{\partial }^2\ell }{\partial {\alpha }_1\partial {\gamma }_2}\\ -\frac{{\partial }^2\ell }{\partial {\alpha }_2\partial {\alpha }_1}& -\frac{{\partial }^2\ell }{\partial {\alpha }_2^2}& -\frac{{\partial }^2\ell }{\partial {\alpha }_2\partial {\theta }_1}& -\frac{{\partial }^2\ell }{\partial {\alpha }_2\partial {\theta }_2}& -\frac{{\partial }^2\ell }{\partial {\alpha }_2\partial {\gamma }_1}& -\frac{{\partial }^2\ell }{\partial {\alpha }_2\partial {\gamma }_2}\\ -\frac{{\partial }^2\ell }{\partial {\theta }_1\partial {\alpha }_1}& -\frac{{\partial }^2\ell }{\partial {\theta }_1\partial {\alpha }_2}& -\frac{{\partial }^2\ell }{\partial {\theta }_1^2}& -\frac{{\partial }^2\ell }{\partial {\theta }_1\partial {\theta }_2}& -\frac{{\partial }^2\ell }{\partial {\theta }_1\partial {\gamma }_1}& -\frac{{\partial }^2\ell }{\partial {\theta }_1\partial {\gamma }_2}\\ -\frac{{\partial }^2\ell }{\partial {\theta }_2\partial {\alpha }_1}& -\frac{{\partial }^2\ell }{\partial {\theta }_2\partial {\alpha }_2}& -\frac{{\partial }^2\ell }{\partial {\theta }_1\partial {\theta }_2}& -\frac{{\partial }^2\ell }{\partial {\theta }_2^2}& -\frac{{\partial }^2\ell }{\partial {\theta }_2\partial {\gamma }_1}& -\frac{{\partial }^2\ell }{\partial {\theta }_2\partial {\gamma }_2}\\ -\frac{{\partial }^2\ell }{\partial {\gamma }_1\partial {\alpha }_1}& -\frac{{\partial }^2\ell }{\partial {\gamma }_1\partial {\alpha }_2}& -\frac{{\partial }^2\ell }{\partial {\gamma }_1\partial {\theta }_1}& -\frac{{\partial }^2\ell }{\partial {\gamma }_1\partial {\theta }_2}& -\frac{{\partial }^2\ell }{\partial {\gamma }_1^2}& -\frac{{\partial }^2\ell }{\partial {\gamma }_1\partial {\gamma }_2}\\ -\frac{{\partial }^2\ell }{\partial {\gamma }_2\partial {\alpha }_1}& -\frac{{\partial }^2\ell }{\partial {\gamma }_2\partial {\alpha }_2}& -\frac{{\partial }^2\ell }{\partial {\gamma }_2\partial {\theta }_1}& -\frac{{\partial }^2\ell }{\partial {\gamma }_2\partial {\theta }_2}& -\frac{{\partial }^2\ell }{\partial {\gamma }_2\partial {\gamma }_1}& -\frac{{\partial }^2\ell }{\partial {\gamma }_2^2}\end{array}\right),\end{array}$$

(32)

hence,

$$\begin{array}{cc}\hfill & I^{-1}({\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,{\gamma }_2)=\hfill \\ & \left(\begin{array}{cccccc}\widehat{var}\left(\widehat{{\alpha }_1}\right)& cov(\widehat{{\alpha }_1},\widehat{{\alpha }_2})& cov(\widehat{{\alpha }_1},\widehat{{\theta }_1})& cov(\widehat{{\alpha }_1},\widehat{{\theta }_2})& cov(\widehat{{\alpha }_1},\widehat{{\gamma }_1})& cov(\widehat{{\alpha }_1},\widehat{{\gamma }_2})\\ cov(\widehat{{\alpha }_2},\widehat{{\alpha }_1})& \widehat{var}\left(\widehat{{\alpha }_2}\right)& cov(\widehat{{\alpha }_2},\widehat{{\theta }_1})& cov(\widehat{{\alpha }_2},\widehat{{\theta }_2})& cov(\widehat{{\alpha }_2},\widehat{{\gamma }_1})& cov(\widehat{{\alpha }_2},\widehat{{\gamma }_2})\\ cov(\widehat{{\theta }_1},\widehat{{\alpha }_1})& cov(\widehat{{\theta }_1},\widehat{{\alpha }_2})& \widehat{var}\left(\widehat{{\theta }_1}\right)& cov(\widehat{{\theta }_1},\widehat{{\theta }_2})& cov(\widehat{{\theta }_1},\widehat{{\gamma }_1})& cov(\widehat{{\theta }_1},\widehat{{\gamma }_2})\\ cov(\widehat{{\theta }_2},\widehat{{\alpha }_1})& cov(\widehat{{\theta }_2},\widehat{{\alpha }_2})& cov(\widehat{{\theta }_2},\widehat{{\theta }_1})& \widehat{var}\left(\widehat{{\theta }_2}\right)& cov(\widehat{{\theta }_2},\widehat{{\gamma }_1})& cov(\widehat{{\theta }_2},\widehat{{\gamma }_2})\\ cov(\widehat{{\gamma }_1},\widehat{{\alpha }_1})& cov(\widehat{{\gamma }_1},\widehat{{\alpha }_2})& cov(\widehat{{\gamma }_1},\widehat{{\theta }_1})& cov(\widehat{{\gamma }_1},\widehat{{\theta }_2})& \widehat{var}\left(\widehat{{\gamma }_1}\right)& cov(\widehat{{\gamma }_1},\widehat{{\gamma }_2})\\ cov(\widehat{{\gamma }_2},\widehat{{\alpha }_1})& cov(\widehat{{\gamma }_2},\widehat{{\alpha }_2})& cov(\widehat{{\gamma }_2},\widehat{{\theta }_1})& cov(\widehat{{\gamma }_2},\widehat{{\theta }_2})& cov(\widehat{{\gamma }_2},\widehat{{\gamma }_1})& \widehat{var}\left(\widehat{{\gamma }_2}\right)\end{array}\right).\hfill \end{array}$$

(33)

From the likelihood function in (9), we have $I_{i,j}\left({\delta }_i\right)=0$ if $i\ne j$. Consequently, we have

$$I\left({\widehat{\delta }}_i\right)=-\mathrm{diagonal}\left(\frac{{\partial }^2\ell }{\partial {\delta }_i^2}{|}_{{\delta }_i={\widehat{\delta }}_i}\right).$$

(34)

The asymptotic normality of the MLEs can be used to compute the ACIs for parameter ${\delta }_i$, so that we can express the approximate $(1-\eta )100\%$ ACIs for ${\delta }_i$ as

$$\begin{array}{c}\hfill \left(\widehat{{\delta }_i}\pm Z_{\eta /2}\sqrt{\widehat{var}\left(\widehat{{\delta }_i}\right)}\right),\end{array}$$

(35)

where $Z_{\eta /2}$ is the percentile of the standard normal distribution with right-tail probability $\eta /2$.

4. Parametric Bootstrap

Bootstrap confidence intervals are a powerful statistical tool used to estimate the uncertainty associated with a parameter or statistic of interest in data analysis. They are particularly valuable when the underlying probability distribution of the data is complex or unknown. The bootstrap technique involves repeatedly resampling the observed data, with replacement, to create a large number of “bootstrap samples.” From these samples, new estimates of the parameter of interest are calculated. By analyzing the distribution of these estimates, bootstrap confidence intervals provide a range within which the true parameter value is likely to fall. This approach is widely used in various fields, such as finance, biology, and machine learning, to gain insight into the robustness and stability of statistical estimates, making it an indispensable tool for researchers and analysts seeking to make informed decisions in the presence of uncertainty.

In this context, we have the Boot-P, introduced by Efron [27], and the Boot-T proposed by Hall [28], along with two variations, Boot-BC and Boot-BCa, which are rooted in the concept presented by DiCiccio and Efron [29]. In 2017, Ghazal and Hasaballah [30] explored bootstrap and MCMC methods based on unified hybrid censored data. Below, we will introduce these four methods.

4.1. Boot-P Method

Use Boot-P when you need a quick and simple estimation of confidence intervals and do not have concerns about bias correction.

• Strengths:
  Simplicity: Boot-P is straightforward to implement.
  Intuitive: It provides easily interpretable confidence intervals based on percentiles.
• Weaknesses:
  May be imprecise for small sample sizes or skewed data. Ignores bias in the original estimator. The steps are given in Algorithm 1.

Algorithm 1: Boot-P method

Depending on the original sample, $\underline{\lambda }=({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n)$ compute the MLEs of the parameters ${\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1\mathrm{and}{\gamma }_2$ from Equations (10)–(15). Utilizing the estimated values ${\widehat{\alpha }}_1,{\widehat{\alpha }}_2,{\widehat{\theta }}_1,{\widehat{\theta }}_2,{\widehat{\gamma }}_1,$ and ${\widehat{\gamma }}_2$, we generate a bootstrap sample denoted as ${\lambda }^{*}$. This sample is constructed to have identical values for $R_i$, where $i=1,2,\dots ,r$. Obtain a bootstrap sample $\underline{{\lambda }^{*}}=({\lambda }_1^{*},{\lambda }_2^{*},\dots ,{\lambda }_n^{*})$ by resampling with replacement. As in Step 1, based on $\underline{{\lambda }^{*}}$, compute the bootstrap sample estimates of $\widehat{\phi }$ where $\widehat{\phi }=[{\widehat{\alpha }}_1,{\widehat{\alpha }}_2,{\widehat{\theta }}_1,{\widehat{\theta }}_2,{\widehat{\gamma }}_1,{\widehat{\gamma }}_2]$, say ${\widehat{\phi }}^{*}=\left[{\widehat{\alpha }}_1^{*},{\widehat{\alpha }}_2^{*},{\widehat{\theta }}_1^{*},{\widehat{\theta }}_2^{*},{\widehat{\gamma }}_1^{*},{\widehat{\gamma }}_2^{*}\right]$. Repeat Steps 3 and 4 N Boot times, and obtain ${\widehat{\phi }}_1^{*},{\widehat{\phi }}_2^{*},\dots .,{\widehat{\phi }}_{NBoot}^{*}$. Sort ${\widehat{{\phi }_i}}^{*},i=1,2,\dots .,NBoot$ in ascending order to obtain the bootstrap sample $\left({\widehat{\phi }}_{\left(1\right)}^{*},{\widehat{\phi }}_{\left(2\right)}^{*},\dots .,{\widehat{\phi }}_{\left(NBoot\right)}^{*}\right)$. Let $G_1\left(z\right)=p({\widehat{\phi }}_i^{*}\le z)$ be the CDF of ${\widehat{\phi }}_i^{*}$. Define ${\widehat{\phi }}_{iBoot}^{*}=G_1^{-1}\left(z\right)$ for the given z. The approximate Boot-p $100(1-\eta )\%$ CI of $\widehat{\phi }$ is given by $\left[{\phi }_{iBoot-p}^{*}\left(\frac{\eta }{2}\right),{\phi }_{iBoot-p}^{*}(1-\frac{\eta }{2})\right]$.

4.2. Boot-T Method

Use Boot-T when you have a small sample size or suspect that the distribution of the statistic is not symmetric.

• Strengths:
  More robust for small sample sizes compared to Boot-P.
  Provides improved accuracy when the distribution of the statistic is not symmetric.
• Weaknesses:
  Somewhat more complex than Boot-P.
  May still have bias, although less than Boot-P. The steps are given in Algorithm 2.

Algorithm 2: Boot-T method

Using the same steps (1) to (4) as in the same steps Boot-P. Determine the $T^{*\phi }$ statistic defined as follows: $T^{*\phi }=\frac{\sqrt{N}({\widehat{\phi }}^{*}-\widehat{\phi })}{\sqrt{\widehat{var}\left({\widehat{\phi }}^{*}\right)}}$, where $var\left({\widehat{\phi }}^{*}\right)$ are obtained by using FIM. Repeat Step 1 and 2 NBoot times and obtain $T_1^{*\phi },T_2^{*\phi },\dots .,T_{NBoot}^{*\phi }$. Sort $T_1^{*\phi },T_2^{*\phi },\dots .,T_{NBoot}^{*\phi }$, in ascending order and obtain the ordered sequences $\left(T_{\left(1\right)}^{*\phi },T_{\left(2\right)}^{*\phi },\dots .,T_{\left(NBoot\right)}^{*\phi }\right)$. Let $G_2\left(z\right)=p(T^{*}\le z)$ be the CDF of $T^{*}$. For a given z, define ${\widehat{\phi }}_{Boot-t}\left(z\right)=\widehat{\phi }+N^{-\frac{1}{2}}\sqrt{\widehat{var}\left({\widehat{\phi }}^{*}\right)}{G}_2^{-1}\left(z\right)$. Then, the approximate Boot-t $100(1-\eta )\%$ CI of $\widehat{\phi }$ is given by $\left[{\widehat{\phi }}_{Boot-t}\left(\frac{\eta }{2}\right),{\widehat{\phi }}_{Boot-t}(1-\frac{\eta }{2})\right]$.

4.3. Boot-BC Method

Use Boot-BC when you want to correct for bias in the estimator and have a reasonably sized dataset.

• Strengths:
  Corrects for bias in the original estimator.
  Provides accurate results for many cases.
• Weaknesses:
  Can be computationally intensive.
  May not work well when dealing with extreme outliers or extremely skewed data. The steps are given in Algorithm 3.

Algorithm 3: Boot-BC method

Follow the same procedures (1) to (4) as outlined in the Boot-P method. Compute the probability denoted as $p_0$ based on the ordered bootstrap distribution of ${\widehat{\phi }}^{*}$, which is given by $$p_0=P\left({\widehat{\phi }}^{*}⩽\widehat{\phi }\right)=\frac{\left(\widehat{\phi }{i}^{*}⩽\widehat{\phi }\right)}{N},\mathrm{where}i=1,2,\dots ,N\mathrm{Boot}.$$ (36) Define $\chi$ and ${\chi }^{-1}$ as the CDF and inverse CDF of the standard normal variable z, respectively. Then, the bias-correction constant $z_0$ is given by $$z_0={\chi }^{-1}\left(p_0\right)={\chi }^{-1}\left(\frac{\left(\widehat{\phi }{i}^{*}⩽\widehat{\phi }\right)}{N}\right),\mathrm{where}i=1,2,\dots ,N\mathrm{Boot}.$$ (37) Here, $P(\widehat{\phi }{i}^{*}⩽\widehat{\phi })=G\left(z_0\right)$, where $G(.)$ is the CDF of the bootstrap distribution. The percentiles of the ordered bootstrap distribution of $\widehat{\phi }$ are computed as $$L=\chi (2z_0+z\eta /2),U=\chi (2z_0+z_{1-\eta /2}).$$ (38) Finally, the approximate Boot-BC $100(1-\eta )\%$ CI for $\widehat{\phi }$ is given by $\left[\widehat{\phi }\mathrm{Boot}-\mathrm{BC}\left(\frac{\eta }{2}\right),\widehat{\phi }\mathrm{Boot}-\mathrm{BC}(1-\frac{\eta }{2})\right]$.

4.4. Boot-BCa Method

Use Boot-BCa when you have a relatively large dataset and need to correct for both bias and skewness.

• Strengths:
  Addresses both bias and skewness in the original estimator. Offers improved accuracy over Boot-BC.
• Weaknesses:
  Even more computationally demanding than Boot-BC.
  Requires larger sample sizes than Boot-BC to perform well. The steps are given in Algorithm 4.

Algorithm 4: Boot-BCa method

Follow the same steps (1) to (4) as outlined in the Boot-P method. Define $\chi \left(z\right)=\vartheta$ as the CDF of the standard normal, where $z_{\vartheta }={\chi }^{-1}\left(\vartheta \right)$, and introduce the bias-correction constant $z_0$ defined in (37). Then, calculate ${\widehat{\phi }}_{\mathrm{Boot}-\mathrm{BCa}}^{*}$ as follows: $${\widehat{\phi }}_{\mathrm{Boot}-\mathrm{BCa}}^{*}=G^{-1}\left[\chi \left(z_0+\frac{z_0+z\vartheta }{1-e(z_0+z_{\vartheta })}\right)\right],$$ (39) where e represents the acceleration factor, and its value can be computed using a jackknife approach. Express the acceleration factor e as $$e={\displaystyle \frac{{\sum }_{i=1}^N{\left(\overline{\phi }-\widehat{\phi }i\right)}^3}{6{\left[\sum {i=1}^N{\left(\overline{\phi }-\widehat{\phi }i\right)}^2\right]}^{\frac{3}{2}}}},\mathrm{where}i=1,2,\dots ,N\mathrm{Boot}.$$ (40) This method has been studied by various authors, including Efron and Tibshirani [31]. Then, the approximate Boot-BCa $100(1-\eta )\%$ CI for $\widehat{\phi }$ is given by $\left[\widehat{\phi }\mathrm{Boot}-\mathrm{BCa}\left(\frac{\eta }{2}\right),\widehat{\phi }\mathrm{Boot}-\mathrm{BCa}(1-\frac{\eta }{2})\right]$.

In summary, the choice of bootstrap method depends on your specific data and research objectives. Boot-P is the simplest but may be less accurate for small samples and skewed data. Boot-T is a better choice for small samples and non-symmetric distributions. Boot-BC and Boot-BCa are appropriate when you want to correct for bias, with Boot-BCa providing more comprehensive corrections but at a higher computational cost. Consider the trade-offs between computational complexity and the need for bias correction when selecting the appropriate bootstrap method for your analysis.

5. Bayesian Estimation

In this section, we employ the MCMC approach within a Gibbs sampler framework that incorporates a nested M-H algorithm. We utilize this approach to generate parametric samples representing the unknown parameters ${\alpha }_1$, ${\alpha }_2$, ${\theta }_1$, ${\theta }_2$, ${\gamma }_1$ and ${\gamma }_2$ from their respective marginal posteriors, allowing us to derive Bayes estimates for these parameters. The Gibbs sampler is a recursive sampling technique employed to simulate samples from the full conditional posterior distributions, while the M-H algorithm is used to generate samples from arbitrary distributions (Hastings [32]; Metropolis et al. [33]). In this context, we generate N samples using the MCMC technique, with the initial M values discarded during the burn-in period. The remaining N–M sample values are subsequently utilized for further Bayesian analysis.

We assume that the parameters ${\alpha }_1$, ${\alpha }_2$, ${\theta }_1$, ${\theta }_2$, ${\gamma }_1$ and ${\gamma }_2$ have independent gamma prior distributions as

$${\pi }_i\left({\delta }_i\right)\propto {\delta }_i^{a_i-1}{e}^{-b_i{\delta }_i},$$

(41)

where $i=1,2,3,4,5,6 \mathrm{and} ({\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4,{\delta }_5,{\delta }_6)=({\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,{\gamma }_2).$

Where the hyperparameters $a_i$ and $b_i$, $i=1,2,3,4,5,6$ are supposed to be known and non-negative. By using the prior distribution for ${\alpha }_1$, ${\alpha }_2$, ${\theta }_1$, ${\theta }_2$, ${\gamma }_1$ and ${\gamma }_2$, we obtain the joint prior distribution as follows:

$$\begin{array}{cc}\hfill \pi ({\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,{\gamma }_2)\propto & {\alpha }_1^{a_1-1}{\alpha }_2^{a_2-1}{\theta }_1^{a_3-1}{\theta }_2^{a_4-1}{\gamma }_1^{a_5-1}{\gamma }_2^{a_6-1}\hfill \\ & \times e^{-b_1{\alpha }_1-b_2{\alpha }_2-b_3{\theta }_1-b_4{\theta }_2-b_5{\gamma }_1-b_6{\gamma }_2}.\hfill \end{array}$$

(42)

Based on (8) and (42), the joint posterior density function of ${\alpha }_1$, ${\alpha }_2$, ${\theta }_1$, ${\theta }_2$, ${\gamma }_1$ and ${\gamma }_2$ given $(H,\lambda ,S)$ is written as

$$\begin{array}{cc}\hfill {\pi }^{*}({\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,{\gamma }_2|H,\lambda ,S)\propto & {\alpha }_1^{a_1-m_r-1}{e}^{-b_1{\alpha }_1}{\alpha }_2^{a_2-n_r-1}{e}^{-b_2{\alpha }_2}{\theta }_1^{a_3+m_r-1}{\theta }_2^{a_4+n_r-1}{\gamma }_1^{a_5+m_r-1}\hfill \\ & \times {\gamma }_2^{a_6+n_r-1}{e}^{-b_3{\theta }_1}{e}^{-b_5{\gamma }_1}{e}^{({\theta }_1-1)\left\{{\sum }_{i=1}^r{h}_{i}ln{\lambda }_i-m_{r}ln{\alpha }_1\right\}}\hfill \\ & \times e^{-b_4{\theta }_2}{e}^{-b_6{\gamma }_2}{e}^{({\theta }_2-1)\left\{{\sum }_{i=1}^r(1-h_i)ln{\lambda }_i-n_{r}ln{\alpha }_2\right\}}\hfill \\ & \times e^{-{\sum }_{i=1}^r\left\{\left({\gamma }_1(h_i+s_i)+h_i\right)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_1}\right)}^{{\theta }_1}\right]\right\}}\hfill \\ & \times e^{-{\sum }_{i=1}^r\left\{\left({\gamma }_2(t_i-h_i+1)-h_i+1\right)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_2}\right)}^{{\theta }_2}\right]\right\}}.\hfill \end{array}$$

(43)

We observed that (43) is not amenable to analytical solutions due to the formidable challenge of deriving closed-form expressions for the marginal posterior distributions of individual parameters. Consequently, we recommend the utilization of the MCMC method to approximate (43). Several studies have extensively explored the MCMC technique, including works by Chen and Shao [34] and Ghazal and Hasaballah [30,35,36]. From (43), the conditional posterior density function of ${\alpha }_1$, ${\alpha }_2$, ${\theta }_1$, ${\theta }_2$, ${\gamma }_1$ and ${\gamma }_2$ can be obtained as the following proportionality: to simplify, we used ${\pi }_1^{*}\left({\alpha }_1\right)$, ${\pi }_2^{*}\left({\alpha }_2\right)$, ${\pi }_3^{*}\left({\theta }_1\right)$, ${\pi }_4^{*}\left({\theta }_2\right)$, ${\pi }_5^{*}\left({\gamma }_1\right)$ and ${\pi }_6^{*}\left({\gamma }_2\right)$ instead of ${\pi }_1^{*}\left({\alpha }_1\right|{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,{\gamma }_2,H,\lambda ,S)$, ${\pi }_2^{*}\left({\alpha }_2\right|{\alpha }_1,{\theta }_1,{\theta }_2,{\gamma }_1,{\gamma }_2,H,\lambda ,S)$, ${\pi }_3^{*}\left({\theta }_1\right|{\alpha }_1,{\alpha }_2,{\theta }_2,{\gamma }_1,{\gamma }_2,H,\lambda ,S)$, ${\pi }_4^{*}\left({\theta }_2\right|{\alpha }_1,{\alpha }_2,{\theta }_1,{\gamma }_1,{\gamma }_2,H,\lambda ,S)$, ${\pi }_5^{*}\left({\gamma }_1\right|{\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_2,H,\lambda ,S)$ and ${\pi }_6^{*}\left({\gamma }_2\right|{\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,H,\lambda ,S),$ respectively:

$$\begin{array}{cc}\hfill {\pi }_1^{*}\left({\alpha }_1\right)\propto & {\alpha }_1^{a_1-m_r-1}{e}^{-[b_1{\alpha }_1+({\theta }_1-1)m_{r}ln{\alpha }_1]}\hfill \\ & \times e^{-{\sum }_{i=1}^r\left\{\left({\gamma }_1(h_i+s_i)+h_i\right)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_1}\right)}^{{\theta }_1}\right]\right\}},\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill {\pi }_2^{*}\left({\alpha }_2\right)\propto & {\alpha }_2^{a_2-n_r-1}{e}^{-[b_2{\alpha }_2+({\theta }_2-1)n_{r}ln{\alpha }_2]}\hfill \\ & \times e^{-{\sum }_{i=1}^r\left\{\left({\gamma }_2(t_i-h_i+1)-h_i+1\right)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_2}\right)}^{{\theta }_2}\right]\right\}},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\pi }_3^{*}\left({\theta }_1\right)\propto & {\theta }_1^{a_3+m_r-1}{e}^{-{\theta }_1\left\{b_3-{\sum }_{i=1}^r{h}_{i}ln\left({\lambda }_i\right)+m_{r}ln{\alpha }_1\right\}}\hfill \\ & \times e^{-{\sum }_{i=1}^r\left\{\left({\gamma }_1(h_i+s_i)+h_i\right)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_1}\right)}^{{\theta }_1}\right]\right\}},\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\pi }_4^{*}\left({\theta }_2\right)\propto & {\theta }_2^{a_4+n_r-1}{e}^{-{\theta }_2\left\{b_4-{\sum }_{i=1}^r(1-h_i)ln\left({\lambda }_i\right)+n_{r}ln{\alpha }_2\right\}}\hfill \\ & \times e^{-{\sum }_{i=1}^r\left\{\left({\gamma }_2(t_i-h_i+1)-h_i+1\right)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_2}\right)}^{{\theta }_2}\right]\right\}},\hfill \end{array}$$

(47)

$${\pi }_5^{*}\left({\gamma }_1\right)\propto {\gamma }_1^{a_5+m_r-1}{e}^{-{\gamma }_1\left\{b_5+{\sum }_{i=1}^r(h_i+s_i)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_1}\right)}^{{\theta }_1}\right]\right\}},$$

(48)

$${\pi }_6^{*}\left({\gamma }_2\right)\propto {\gamma }_2^{a_6+n_r-1}{e}^{-{\gamma }_2\left\{b_6+{\sum }_{i=1}^r(t_i-h_i+1)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_2}\right)}^{{\theta }_2}\right]\right\}}.$$

(49)

It is evident that the full conditional posterior density function of ${\gamma }_1$, as provided in Equation (48), takes the form of a gamma density with a shape parameter of $(a_5+m_r)$ and a scale parameter of $\left(a_5+m_r,b_5+{\sum }_{i=1}^r(h_i+s_i)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_1}\right)}^{{\theta }_1}\right]\right)$. Similarly, the full conditional posterior density function of ${\gamma }_2$, as shown in Equation (49), follows a gamma distribution with a shape parameter of $(a_6+n_r)$ and a scale parameter of $\left(a_6+n_r,b_6+{\sum }_{i=1}^r(t_i-h_i+1)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_2}\right)}^{{\theta }_2}\right]\right)$. Consequently, samples of ${\gamma }_1$ and ${\gamma }_2$ can be readily generated using any gamma distribution generation method.

However, it is important to note that the conditional posterior distribution functions of ${\alpha }_1$, ${\alpha }_2$, ${\theta }_1$, and ${\theta }_2$, as described in Equations (44)–(47), cannot be analytically simplified into well-known distributions. Consequently, direct sampling using standard methods can be challenging. Nevertheless, as depicted in Figure 1, these distributions exhibit similarities to the normal distribution.

5.1. Estimation Based on SE Loss Function

The SE loss function is represented by the equation:

$${\xi }_{SE}(\Delta )=a{\Delta }^2=a{[u\left(\theta \right)-\widehat{u}\left(\theta \right)]}^2.$$

(50)

In this equation, the positive constant a is typically set to 1, $\mathsf{\Delta }=\widehat{u}\left(\theta \right)-u\left(\theta \right)$, $u\left(\theta \right)$ is a function of $\theta$ to be estimated, and $\widehat{u}\left(\theta \right)$ denotes the SE estimate of $u\left(\theta \right)$. The Bayes estimator under a quadratic loss function is computed as the mean of the posterior distribution:

$$\widehat{u}\left(\theta \right)SE=E\left[u\left(\theta \right)\right|\underline{x}]=\int \Theta u\left(\theta \right){\pi }^{*}\left(\theta \right|\underline{x})d\theta .$$

(51)

The SE loss function is commonly employed in the literature and is considered one of the most prevalent loss functions. It exhibits symmetry, implying that it treats the overestimation and underestimation of parameters equally. However, in life-testing scenarios, one type of estimation error may have more significant consequences than the other.

5.2. Estimation Based on LINEX Loss Function

The LINEX loss function is defined as follows:

$${\xi }_{LINEX}\left(\mathsf{\Delta }\right)\propto e^{a\mathsf{\Delta }}-a\mathsf{\Delta }-1,a\ne 0.$$

(52)

Here, $\mathsf{\Delta }$ is as previously defined, and $\widehat{u}\left(\theta \right)$ represents the LINEX estimate of $u\left(\theta \right)$.

The shape parameter a determines the direction and degree of symmetry for this loss function. Varian [37] first introduced this loss function, while Zellner [38] highlighted its intriguing properties. When $a>0$, overestimation is penalized more severely than underestimation, and the reverse is true when a is negative. However, for values of a close to zero, the LINEX loss function closely resembles the symmetry of the SE loss function. This function exhibits significant asymmetry when $a=1$ with overestimation incurring higher costs than underestimation. Conversely, when $a<0$, the loss function increases nearly exponentially for $\mathsf{\Delta }=\widehat{u}\left(\theta \right)-u\left(\theta \right)<0$ and decreases nearly linearly for $\mathsf{\Delta }=\widehat{u}\left(\theta \right)-u\left(\theta \right)>0$.

This is applicable provided that $E\left(e^{-au\left(\theta \right)}\right|\underline{x})$ exists and is finite. Now, we will outline the steps of the process for the M-H within Gibbs sampling method in Algorithm 5.

Algorithm 5: Metropolis –Hasting within Gibbs sampling

Begin with initial guess of $\left({\alpha }_1^{\left(0\right)},{\alpha }_2^{\left(0\right)},{\theta }_1^{\left(0\right)},{\theta }_2^{\left(0\right)},{\gamma }_1^{\left(0\right)},{\gamma }_2^{\left(0\right)}\right)$, M = burn-in. Put $j=1$. Generate ${\gamma }_1^{\left(j\right)}$ from Gamma $(a_5+m_r,b_5+{\sum }_{i=1}^r(h_i+s_i)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_1}\right)}^{{\theta }_1}\right]$. Generate ${\gamma }_2^{\left(j\right)}$ from Gamma $(a_6+n_r,b_6+{\sum }_{i=1}^r(t_i-h_i+1)ln\left[1+{\left(\frac{{\lambda }_i}{{\alpha }_2}\right)}^{{\theta }_2}\right]$. Using M-H, generate ${\alpha }_1^{\left(j\right)}$, ${\alpha }_2^{\left(j\right)}$, ${\theta }_1^{\left(j\right)}$ and ${\theta }_2^{\left(j\right)}$ from (44)–(47) with normal suggested distribution $N({\alpha }_1^{(j-1)},var\left({\alpha }_1\right))$, $N({\alpha }_2^{(j-1)},var\left({\alpha }_2\right))$, $N({\theta }_1^{(j-1)},var\left({\theta }_1\right))$ and $N({\theta }_2^{(j-1)},var\left({\theta }_2\right))$ where $var\left({\alpha }_1\right)$, $var\left({\alpha }_2\right)$, $var\left({\theta }_1\right)$ and $var\left({\theta }_2\right)$ can be obtained from the main diagonal in inverse FIM. (i) Generate a proposal ${\alpha }_1^{*}$ from $N({\alpha }_1^{(j-1)},var\left({\alpha }_1\right))$, ${\alpha }_2^{*}$ from $N({\alpha }_2^{(j-1)},var\left({\alpha }_2\right))$, ${\theta }_1^{*}$ from $N({\theta }_1^{(j-1)},var\left({\theta }_1\right))$ and ${\theta }_2^{*}$ from $N({\theta }_2^{(j-1)},var\left({\theta }_2\right))$. (ii) Evaluate the acceptance probabilities $$r_1=\min \left[1,\frac{{\pi }_1^{*}\left({\alpha }_1^{*}\right|{\alpha }_2^{(j-1)},{\theta }_1^{(j-1)},{\theta }_2^{(j-1)},{\gamma }_1^j,{\gamma }_2^j,\underline{x})}{{\pi }_1^{*}\left({\alpha }_1^{(j-1)}\right|{\alpha }_2^{(j-1)},{\theta }_1^{(j-1)},{\theta }_2^{(j-1)},{\gamma }_1^j,{\gamma }_2^j,\underline{x})}\right],$$ (53) $$r_2=\min \left[1,\frac{{\pi }_2^{*}\left({\alpha }_2^{*}\right|{\alpha }_1^j,{\theta }_1^{(j-1)},{\theta }_2^{(j-1)},{\gamma }_1^j,{\gamma }_2^j,\underline{x})}{{\pi }_2^{*}\left({\alpha }_2^{(j-1)}\right|{\alpha }_1^j,{\theta }_1^{(j-1)},{\theta }_2^{(j-1)},{\gamma }_1^j,{\gamma }_2^j,\underline{x})}\right],$$ (54) $$r_3=\min \left[1,\frac{{\pi }_3^{*}\left({\theta }_1^{*}\right|{\alpha }_1^j,{\alpha }_2^j,{\theta }_2^{(j-1)},{\gamma }_1^j,{\gamma }_2^j,\underline{x})}{{\pi }_3^{*}\left({\theta }_1^{(j-1)}\right|{\alpha }_1^j,{\alpha }_2^j,{\theta }_2^{(j-1)},{\gamma }_1^j,{\gamma }_2^j,\underline{x})}\right],$$ (55) $$r_4=\min \left[1,\frac{{\pi }_4^{*}\left({\theta }_2^{*}\right|{\alpha }_1^j,{\alpha }_2^j,{\theta }_1^j,{\gamma }_1^j,{\gamma }_2^j,\underline{x})}{{\pi }_4^{*}\left({\theta }_2^{(j-1)}\right|{\alpha }_1^j,{\alpha }_2^j,{\theta }_1^j,{\gamma }_1^j,{\gamma }_2^j,\underline{x})}\right],$$ (56) (iii) Generate u from a uniform $(0,1)$ distribution. (iv) If $u\le r_1$, do not reject the proposal and put ${\alpha }_1^{\left(j\right)}={\alpha }_1^{*}$; otherwise, put ${\alpha }_1^{\left(j\right)}={\alpha }_1^{(j-1)}$. (v) If $u\le r_2$, do not reject the proposal and put ${\alpha }_2^{\left(j\right)}={\alpha }_2^{*}$; otherwise, put ${\alpha }_2^{\left(j\right)}={\alpha }_2^{(j-1)}$. (vi) If $u\le r_3$, do not reject the proposal and put ${\theta }_1^{\left(j\right)}={\theta }_1^{*}$; otherwise, put ${\theta }_1^{\left(j\right)}={\theta }_1^{(j-1)}$. (vii) If $u\le r_4$, do not reject the proposal and put ${\theta }_2^{\left(j\right)}={\theta }_2^{*}$; otherwise, put ${\theta }_2^{\left(j\right)}={\theta }_2^{(j-1)}$. Put $j=j+1$. Repeat Steps $3-6$ N times and obtain ${\alpha }_1^{\left(j\right)},{\alpha }_2^{\left(j\right)},{\theta }_1^{\left(j\right)},{\theta }_2^{\left(j\right)},{\gamma }_1^{\left(j\right)},$ and ${\gamma }_2^{\left(j\right)}$, $j=M+1,\dots ,N$. To evaluate the CRIs of ${\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1$ and ${\gamma }_2$, order ${\alpha }_1^{\left(j\right)},{\alpha }_2^{\left(j\right)},{\theta }_1^{\left(j\right)},{\theta }_2^{\left(j\right)},{\gamma }_1^{\left(j\right)},$ and ${\gamma }_2^{\left(j\right)}$, $j=M+1,\dots ,N$, as $\left({\alpha }_1^{\left(1\right)}<\dots ,<{\alpha }_1^{\left(N\right)}\right)$, $\left({\alpha }_2^{\left(1\right)}<\dots ,<{\alpha }_2^{\left(N\right)}\right)$, $\left({\theta }_1^{\left(1\right)}<\dots ,<{\theta }_1^{\left(N\right)}\right)$, $\left({\theta }_2^{\left(1\right)}<\dots ,<{\theta }_2^{\left(N\right)}\right)$, $\left({\gamma }_1^{\left(1\right)}<\dots ,<{\gamma }_1^{\left(N\right)}\right)$ and $\left({\gamma }_2^{\left(1\right)}<\dots ,<{\gamma }_2^{\left(N\right)}\right)$. Then, the $100(1-\eta )\%$ CRIs of $\psi ={\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1$ or ${\gamma }_2$ is $\left({\psi }_{{\left(N\right)}_{\eta /2}},{\psi }_{{\left(N\right)}_{(1-\eta /2)}}\right)$. Then, we obtain the Bayes estimates of $\psi$ based on SE loss function as $${\widehat{\psi }}_{BS}=E\left[\psi \right|\underline{x}]=\frac{1}{N-M}\sum _{i=M+1}^N{\psi }^{\left(j\right)},$$ (57) and the Bayes estimates of $\psi$ based on LINEX loss function as $${\widehat{\psi }}_{BL}=-\frac{1}{a}ln\left[\frac{1}{N-M}\sum _{i=M+1}^N{e}^{-a{\psi }^{\left(j\right)}}\right].$$ (58)

The posterior expectation of the LINEX loss function (52) can be expressed as follows:

$$E\left[{\xi }_{LINEX}[\widehat{u}\left(\theta \right)-u\left(\theta \right)]|\mathbf{x}\right]\propto e^{a\widehat{u}\left(\theta \right)}E\left[e^{-au\left(\theta \right)}\right|\underline{x}]-a(\widehat{u}\left(\theta \right)-E\left[u\left(\theta \right)\right|\underline{x}])-1.$$

(59)

Using the LINEX loss function, the Bayes estimate of $u\left(\theta \right)$ is given by the following:

$$\widehat{u}{\left(\theta \right)}_{LINEX}=\frac{-1}{a}ln\left[E\left(e^{-au\left(\theta \right)}\right|\underline{x})\right].$$

(60)

6. Applications

In this part, we examine an actual set of data to illustrate how the suggested techniques operate in practical situations. The dataset we used was initially obtained from the National Climatic Data Center (NCDC) located in Asheville, North Carolina, USA. These data show the wind speed measured in knots for two samples: the first one for 23 days and the second one for 25 days. We calculated the average wind speeds for Alexandria city on a daily basis from 1 February 2017 to 23 February 2017 and from 1 February 2018 to 25 February 2018, respectively, in

Table 1. For the first sample.

8.6	3.8	5.4	4.4	2.2	3.8	4.5	6.3	3.4	4.1	3.8	8.6	13.0	11.3
12.4	12.4	5.0	3.4	3.8	5.3	3.6	5.8	4.2

and

Table 2. For the second sample.

2.4	2.9	3.3	3.4	3.5	3.7	3.8	3.9	4.0	4.1	4.2	4.5	4.6	4.8	5.1
5.3	5.5	6.0	6.2	6.5	7.8	8.2	8.4	9.4	10.9

as follows:

We used the Kolmogorov–Smirnov (K-S) test to check if the data distribution fit the TPBXIID model. For the first sample, the K-S test calculated a value of $0.114201$ for TPBXIID, which is smaller than the expected value of $0.2749$ at a significance level of $5\%$ with $n=23$ and a P-value of $0.892195$. Similarly, for the second sample, the K-S test calculated a value of $0.0825997$ for TPBXIID, which is smaller than the expected value of $0.2640$ at a significance level of $5\%$ with $n=25$ and a P-value of $0.990103$. Therefore, we can conclude that TPBXIID fits both samples very well. We have also included Figure 2 and Figure 3 to show how well the empirical and fitted values match up. Overall, TPBXIID seems to be an excellent model for fitting these data.

From the above datasets, we have generated the JP-II-C sample with the censoring scheme. Assume that $m=23$ for the first sample and $n=25$ for the second sample; by implementing JP-II-CS where $K=m+n$ denotes the total sample size and when $r=10$, $S=(5,0,0,0,5,0,0,0,0,8)$ and $T=(5,0,0,0,5,0,0,0,0,10)$, then $R=(10,0,0,0,10,0,0,0,0,18)$.

The generated datasets are provided below.

$$\lambda =(2.2,3.3,3.4,3.4,3.5,3.6,3.7,3.8,3.8,3.8)\mathrm{and}H=(1,0,1,1,0,1,0,1,1,1).$$

Depending on the data type used in this study, we calculate estimates based on MLEs and the bootstrap method for ${\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1$ and ${\gamma }_2$; the results are shown in

Table 3. Different point estimates of $({\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,{\gamma }_2)$.

Parameters	MLE	Boot-P	Boot-T	MCMC
				SE	LINEX
					a = −3.0	a = 0.0001	a = 3.0
${\alpha }_1$	4.1568	4.2619	3.9935	4.1499	4.1499	4.1499	4.1498
${\alpha }_2$	3.9003	4.4855	3.1616	4.1739	4.1739	4.1928	4.1548
${\theta }_1$	8.7082	9.5529	7.9553	8.8083	8.8083	8.8159	8.8006
${\theta }_2$	5.692	4.678	5.0617	5.7879	5.7879	5.8103	5.7670
${\gamma }_1$	1.1693	1.6072	0.8190	0.7537	0.7536	0.9386	0.6400
${\gamma }_2$	0.4261	0.8223	0.4000	0.4168	0.4168	0.5036	0.3620

, and the results of $95\%$ ACIs, Boot-P CI, Boot-T CI, Boot-BC CI and Boot-BCa CI for ${\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1$ and ${\gamma }_2$ are given in

Table 4. The 95% CIs/CRIs for $({\alpha }_1,{\alpha }_2)$.

Method	${\mathit{\alpha }}_1$			${\mathit{\alpha }}_2$
	Lower	Upper	Length	Lower	Upper	Length
ACI	2.2536	6.0599	3.80628	−27.9891	35.7898	63.779
Boot-p CI	3.2003	5.6068	2.40653	1.8659	7.0946	5.2287
Boot-t CI	3.6257	4.1222	0.496511	0.0597	3.8215	3.76171
Boot-BC CI	2.1463	4.8118	2.66547	0.8629	5.3733	4.51041
Boot-BCa CI	2.1463	4.8118	2.6654	0.8629	5.1814	4.31844
CRI	4.1396	4.1602	0.0206	3.954	4.3619	0.4079

,

Table 5. The 95% CIs/CRIs for $({\theta }_1,{\theta }_2)$.

Method	${\mathit{\theta }}_1$			${\mathit{\theta }}_2$
	Lower	Upper	Length	Lower	Upper	Length
ACI	2.157	15.2593	13.1023	−26.0556	37.4395	63.4951
Boot-p CI	4.9743	18.7862	13.8119	2.6508	7.5756	4.92486
Boot-t CI	5.8477	8.4844	2.6367	2.7803	5.5361	2.75581
Boot-BC CI	4.8807	17.3566	12.4759	4.3803	7.9951	3.6147
Boot-BCa CI	4.6822	14.4604	9.7782	4.3783	7.9951	3.6167
CRI	8.7025	8.9157	0.2131	5.609	5.9931	0.3841

and

Table 6. The 95% CIs/CRIs for $({\gamma }_1,{\gamma }_2)$.

Method	${\mathit{\gamma }}_1$			${\mathit{\gamma }}_2$
	Lower	Upper	Length	Lower	Upper	Length
ACI	−2.5035	4.8422	7.3457	−13.9921	14.8443	28.8364
Boot-p CI	0.1650	5.5745	5.40953	0.0397	3.0249	2.9852
Boot-t CI	−0.6884	1.1492	1.83756	−0.7224	0.4234	1.1458
Boot-BC CI	0.1117	3.4082	3.2965	0.0267	1.2617	1.2350
Boot-BCa CI	0.0757	2.9723	2.8966	0.0267	1.1314	1.1047
CRI	0.2772	1.4735	1.1963	0.1183	0.9243	0.8060

. We used the MCMC method with a 11000 MCMC sample for Bayesian estimation and ignored the first 1000 values as ‘burn-in’. We also used the non-informative priors with hyperparameters $a_i=0.0$ and $b_i=0.0$. Then, we obtained the Bayesian estimates for ${\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1$ and ${\gamma }_2$ under SE loss and LINEX loss functions, and the results are displayed in Table 3. Moreover, the results of the $95\%$ CRIs for ${\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1$ and ${\gamma }_2$ are tabled in Table 4, Table 5 and Table 6.

7. Simulation Study

In this section, we have conducted a comprehensive simulation study to assess and compare the performance of various methods. Our investigation encompassed a range of sample sizes for two distinct populations, with m and n taking on values of 10, 20, 30, 40, 50, 60, and multiple selections for the JP-II-CS, specifically with r values of 5, 10, 15, 20, 30, 40, 50, 60, 70, and 80. For the purpose of our simulations, we established fixed parameter values, precisely $({\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,{\gamma }_2)=(3.4983,3.7686,8.5885,7.1574,0.2484,0.4110)$, and subsequently calculated MLEs along with 95% CIs for these parameters $({\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1,{\gamma }_2)$ across all the specified scenarios. This process was repeated 1000 times, and we calculated the mean values of MLEs, lengths, and CP. The results are displayed in

Table 7. MSEs, lengths and CPs for parameter ${\alpha }_1$.

$(\mathit{m},\mathit{n})$	r	(${\mathit{R}}_1,\mathit{\dots },{\mathit{R}}_{\mathit{r}}$)	MLE			MCMC
			MSE	Length	CP	SE	LINEX		Length	CP
							a = −3	a = 3
(10, 10)	5	$(0_{\left(4\right)},15)$	0.0242	6.820	0.854	0.0239	0.0239	0.0237	0.0546	0.902
		$(0_{\left(3\right)},2,13)$	0.0214	9.293	0.873	0.0210	0.0209	0.0212	0.0884	0.915
	10	$(0_{\left(9\right)},10)$	0.0279	13.7368	0.864	0.0214	0.0217	0.0212	0.1134	0.905
		$(10,0_{\left(9\right)})$	0.0126	5.7552	0.897	0.0125	0.0125	0.0123	0.0434	0.926
	15	$(0_{\left(14\right)},5)$	0.0280	7.0337	0.886	0.0255	0.0256	0.0254	0.0678	0.901
		$(5,0_{\left(14\right)})$	0.0311	4.5501	0.865	0.0305	0.0305	0.0304	0.0409	0.904
(10, 20)	10	$(0_{\left(9\right)},20)$	0.0044	10.3981	0.903	0.0041	0.0042	0.0041	0.0693	0.928
		$(20,0_{\left(9\right)})$	0.0006	5.8821	0.914	0.0006	0.0006	0.0005	0.0502	0.956
	15	$(0_{\left(14\right)},15)$	0.0038	6.8529	0.907	0.0037	0.0037	0.0035	0.0563	0.929
		$(15,0_{\left(14\right)})$	0.0190	5.5755	0.904	0.0109	0.0109	0.0106	0.0433	0.938
	20	$(0_{\left(19\right)},10)$	0.0551	8.321	0.873	0.0501	0.0503	0.050	0.0739	0.907
		$(10,0_{\left(19\right)})$	0.0194	5.0308	0.881	0.0192	0.0192	0.0190	0.0369	0.944
(20, 10)	10	$(0_{\left(9\right)},20)$	0.0035	5.9895	0.903	0.0035	0.0035	0.0033	0.0515	0.945
		$(0_{\left(8\right)},2,18)$	0.0044	7.2016	0.883	0.0037	0.0038	0.0037	0.0632	0.928
	15	$(0_{\left(14\right)},15)$	0.0006	4.2095	0.907	0.0001	0.0004	0.0002	0.0411	0.956
		$(15,0_{\left(14\right)})$	0.0092	3.9171	0.882	0.0088	0.0088	0.0085	0.0285	0.928
	20	$(0_{\left(19\right)},10)$	0.0248	5.6331	0.857	0.0242	0.0242	0.0241	0.0509	0.914
		$(10,0_{\left(19\right)})$	0.0111	3.1076	0.863	0.0109	0.0109	0.0104	0.0245	0.937
(20, 30)	20	$(0_{\left(19\right)},30)$	0.0002	6.5341	0.924	0.0001	0.0001	0.0001	0.0537	0.956
		$(30,0_{\left(19\right)})$	0.0103	4.0279	0.863	0.0098	0.0098	0.0092	0.0361	0.904
	30	$(0_{\left(29\right)},20)$	0.0022	3.8038	0.881	0.0021	0.0021	0.0020	0.0344	0.908
		$(20,0_{\left(29\right)})$	0.0176	3.6818	0.863	0.0167	0.0167	0.0162	0.0275	0.914
	40	$(0_{\left(39\right)},10)$	0.0253	3.8069	0.854	0.0251	0.0251	0.0250	0.0347	0.879
		$(10,0_{\left(39\right)})$	0.0065	2.9572	0.892	0.0061	0.0061	0.0060	0.0233	0.925
(30, 20)	20	$(0_{\left(19\right)},30)$	0.0037	4.7041	0.887	0.0035	0.0035	0.0032	0.0357	0.904
		$(30,0_{\left(19\right)})$	0.0118	3.5749	0.875	0.0110	0.0111	0.0110	0.0304	0.896
	30	$(0_{\left(29\right)},20)$	0.0018	3.148	0.883	0.0016	0.0016	0.0014	0.0264	0.935
		$(20,0_{\left(29\right)})$	0.0071	2.235	0.893	0.0068	0.0068	0.0062	0.0199	0.927
	40	$(0_{\left(39\right)},10)$	0.0116	3.1023	0.854	0.0114	0.0114	0.0112	0.022	0.908
		$(10,0_{\left(39\right)})$	0.0275	2.4304	0.844	0.0267	0.0268	0.0267	0.025	0.895
(40, 50)	40	$(0_{\left(39\right)},50)$	0.0034	5.0679	0.872	0.0032	0.0033	0.0032	0.0466	0.924
		$(50,0_{\left(39\right)})$	0.0089	2.7426	0.884	0.0087	0.0088	0.0086	0.0241	0.912
	50	$(0_{\left(49\right)},40)$	0.0095	3.795	0.882	0.0093	0.0094	0.0092	0.0308	0.934
		$(40,0_{\left(49\right)})$	0.0044	2.7326	0.866	0.0041	0.0042	0.0040	0.0212	0.907
	60	$(0_{\left(59\right)},30)$	0.0223	2.8939	0.886	0.0212	0.0215	0.0211	0.0245	0.919
		$(30,0_{\left(59\right)})$	0.0133	2.5768	0.874	0.0131	0.0132	0.0130	0.0226	0.938
(50, 40)	40	$(0_{\left(39\right)},50)$	0.0004	3.6685	0.881	0.0002	0.0003	0.0001	0.0373	0.954
		$(50,0_{\left(39\right)})$	0.0081	2.2462	0.857	0.0070	0.0072	0.006	0.0172	0.889
	50	$(0_{\left(49\right)},40)$	0.0049	3.3794	0.891	0.0047	0.0048	0.0046	0.0312	0.946
		$(40,0_{\left(49\right)})$	0.0116	2.1702	0.873	0.0114	0.0115	0.0114	0.0178	0.921
	60	$(0_{\left(59\right)},30)$	0.0117	2.2345	0.855	0.0122	0.0125	0.0121	0.0164	0.903
		$(30,0_{\left(59\right)})$	0.0005	1.8105	0.892	0.0002	0.0003	0.0001	0.0168	0.954
(50, 60)	60	$(0_{\left(59\right)},50)$	0.0051	3.3629	0.875	0.0049	0.0050	0.0048	0.0215	0.916
		$(50,0_{\left(59\right)})$	0.0149	2.5738	0.863	0.0147	0.0148	0.0146	0.0200	0.926
	70	$(0_{\left(69\right)},40)$	0.0039	2.478	0.882	0.0037	0.0038	0.0036	0.0197	0.904
		$(40,0_{\left(69\right)})$	0.0155	2.167	0.871	0.0152	0.0153	0.0151	0.0142	0.912
	80	$(0_{\left(79\right)},30)$	0.0026	1.9822	0.881	0.0022	0.0024	0.0020	0.016	0.931
		$(30,0_{\left(79\right)})$	0.0021	1.7481	0.876	0.0018	0.0019	0.0017	0.0133	0.945
(60, 50)	60	$(0_{\left(59\right)},50)$	0.0008	2.3816	0.907	0.0006	0.0007	0.0005	0.0191	0.953
		$(50,0_{\left(59\right)})$	0.0025	2.0017	0.887	0.0023	0.0024	0.0021	0.0135	0.932
	70	$(0_{\left(69\right)},40)$	0.0058	2.5617	0.868	0.0054	0.0055	0.0052	0.0222	0.907
		$(40,0_{\left(69\right)})$	0.0056	1.878	0.894	0.0054	0.0055	0.0053	0.014	0.923
	80	$(0_{\left(79\right)},30)$	0.0080	2.2131	0.872	0.0075	0.0079	0.0072	0.0178	0.916
		$(30,0_{\left(79\right)})$	0.0018	1.3022	0.864	0.0015	0.0016	0.0013	0.0086	0.902

,

Table 8. MSEs, lengths and CPs for parameter ${\alpha }_2$.

$(\mathit{m},\mathit{n})$	r	(${\mathit{R}}_1,\mathit{\dots },{\mathit{R}}_{\mathit{r}}$)	MLE			MCMC
			MSE	Length	CP	SE	LINEX		Length	CP
							a = −3	a = 3
(10, 10)	5	$(0_{\left(4\right)},15)$	0.0115	10.3211	0.852	0.0096	0.0097	0.0095	0.0785	0.904
		$(0_{\left(3\right)},2,13)$	0.0058	9.7619	0.862	0.0054	0.0055	0.0054	0.0807	0.931
	10	$(0_{\left(9\right)},10)$	0.0097	10.3294	0.851	0.0089	0.0090	0.0089	0.0932	0.903
		$(10,0_{\left(9\right)})$	0.0048	5.5611	0.872	0.0043	0.0043	0.0042	0.0456	0.924
	15	$(0_{\left(14\right)},5)$	0.0004	4.429	0.881	0.0001	0.0002	0.0001	0.0392	0.953
		$(5,0_{\left(14\right)})$	0.0016	5.1453	0.882	0.0014	0.0014	0.0012	0.0389	0.941
(10, 20)	10	$(0_{\left(9\right)},20)$	0.0339	6.1937	0.854	0.0338	0.0337	0.0332	0.046	0.902
		$(20,0_{\left(9\right)})$	0.0033	5.431	0.871	0.0031	0.0031	0.0030	0.0438	0.913
	15	$(0_{\left(14\right)},15)$	0.0004	6.5492	0.883	0.0001	0.0002	0.0001	0.0578	0.952
		$(15,0_{\left(14\right)})$	0.0003	3.3861	0.892	0.0001	0.0002	0.0001	0.024	0.954
	20	$(0_{\left(19\right)},10)$	0.0015	3.905	0.884	0.0012	0.0013	0.0011	0.0262	0.941
		$(10,0_{\left(19\right)})$	0.0022	3.8992	0.885	0.0017	0.0018	0.0016	0.0364	0.924
(20, 10)	10	$(0_{\left(9\right)},20)$	0.0759	4.2263	0.876	0.0751	0.0752	0.0750	0.0331	0.903
		$(0_{\left(8\right)},2,18)$	0.0222	9.3283	0.842	0.0204	0.0206	0.0201	0.0989	0.893
	15	$(0_{\left(14\right)},15)$	0.0017	10.456	0.864	0.0012	0.0013	0.0012	0.0924	0.903
		$(15,0_{\left(14\right)})$	0.0166	5.3404	0.872	0.0165	0.0165	0.0164	0.0485	0.914
	20	$(0_{\left(19\right)},10)$	0.0005	5.5859	0.901	0.0001	0.0003	0.0001	0.0488	0.954
		$(10,0_{\left(19\right)})$	0.0018	4.3324	0.882	0.0013	0.0013	0.0012	0.043	0.945
(20, 30)	20	$(0_{\left(19\right)},30)$	0.0005	6.6456	0.882	0.0003	0.0004	0.0002	0.061	0.933
		$(30,0_{\left(19\right)})$	0.0006	3.1585	0.892	0.0003	0.0004	0.0003	0.0209	0.942
	30	$(0_{\left(29\right)},20)$	0.0024	3.217	0.883	0.0022	0.0023	0.0020	0.0246	0.912
		$(20,0_{\left(29\right)})$	0.0035	2.5193	0.871	0.0032	0.0033	0.0031	0.0192	0.924
	40	$(0_{\left(39\right)},10)$	0.0004	2.6279	0.891	0.0002	0.0003	0.0001	0.0185	0.952
		$(10,0_{\left(39\right)})$	0.0062	2.4649	0.881	0.0064	0.0064	0.0064	0.0209	0.933
(30, 20)	20	$(0_{\left(19\right)},30)$	0.0058	6.4601	0.861	0.0054	0.0055	0.0052	0.0429	0.904
		$(30,0_{\left(19\right)})$	0.0006	3.576	0.902	0.0004	0.0005	0.0004	0.0289	0.954
	30	$(0_{\left(29\right)},20)$	0.0056	5.244	0.861	0.0052	0.0054	0.0052	0.0507	0.911
		$(20,0_{\left(29\right)})$	0.0006	3.4225	0.883	0.0003	0.0004	0.0002	0.0316	0.952
	40	$(0_{\left(39\right)},10)$	0.0088	5.0224	0.891	0.0079	0.0080	0.0079	0.0397	0.944
		$(10,0_{\left(39\right)})$	0.0008	2.6781	0.902	0.0005	0.0006	0.0004	0.0226	0.951
(40, 50)	40	$(0_{\left(39\right)},50)$	0.0026	4.4594	0.872	0.0021	0.0022	0.0020	0.0315	0.932
		$(50,0_{\left(39\right)})$	0.019	2.9672	0.874	0.0161	0.0163	0.0160	0.0262	0.904
	50	$(0_{\left(49\right)},40)$	0.0036	4.122	0.881	0.0032	0.0033	0.0031	0.035	0.922
		$(40,0_{\left(49\right)})$	0.0099	2.3561	0.882	0.0095	0.0096	0.0094	0.0176	0.942
	60	$(0_{\left(59\right)},30)$	0.0008	2.1063	0.891	0.0003	0.0004	0.0002	0.0168	0.954
		$(30,0_{\left(59\right)})$	0.0009	1.7113	0.902	0.0005	0.0006	0.0005	0.0115	0.952
(50, 40)	40	$(0_{\left(39\right)},50)$	0.0008	4.2964	0.893	0.0002	0.0004	0.0001	0.0333	0.941
		$(50,0_{\left(39\right)})$	0.0019	2.8549	0.864	0.0016	0.0017	0.0015	0.0199	0.903
	50	$(0_{\left(49\right)},40)$	0.0007	4.3184	0.891	0.0003	0.0004	0.0002	0.0376	0.954
		$(40,0_{\left(49\right)})$	0.0028	2.3438	0.882	0.0024	0.0026	0.0023	0.0184	0.932
	60	$(0_{\left(59\right)},30)$	0.0008	2.7736	0.891	0.0002	0.0005	0.0001	0.0231	0.943
		$(30,0_{\left(59\right)})$	0.0006	2.0112	0.902	0.0003	0.0004	0.0001	0.0156	0.951
(50, 60)	60	$(0_{\left(59\right)},50)$	0.0009	3.1358,	0.903	0.0005	0.0006	0.0004	0.0348	0.954
		$(50,0_{\left(59\right)})$	0.0009	1.5612	0.883	0.0006	0.0007	0.0005	0.0133	0.942
	70	$(0_{\left(69\right)},40)$	0.0005	2.5098	0.891	0.0003	0.00041	0.0001	0.0225	0.952
		$(40,0_{\left(69\right)})$	0.0064	2.2639	0.882	0.0057	0.0059	0.0055	0.0196	0.932
	80	$(0_{\left(79\right)},30)$	0.0019	2.3721	0.852	0.0017	0.0018	0.0016	0.0204	0.901
		$(30,0_{\left(79\right)})$	0.0128	2.2228	0.864	0.0124	0.0126	0.0122	0.0169	0.911
(60, 50)	60	$(0_{\left(59\right)},50)$	0.0078	1.9201	0.844	0.0073	0.0075	0.0071	0.0200	0.891
		$(50,0_{\left(59\right)})$	0.0029	2.532	0.874	0.0026	0.0028	0.0024	0.0185	0.902
	70	$(0_{\left(69\right)},40)$	0.0018	2.6599	0.871	0.0014	0.0015	0.0012	0.0219	0.922
		$(40,0_{\left(69\right)})$	0.0019	2.570	0.871	0.0016	0.0017	0.0015	0.0198	0.912
	80	$(0_{\left(79\right)},30)$	0.0008	1.9509	0.881	0.0004	0.0005	0.0002	0.0159	0.933
		$(30,0_{\left(79\right)})$	0.0007	2.0237	0.893	0.0004	0.0005	0.0002	0.0173	0.954

,

Table 9. MSEs, lengths and CPs for parameter ${\theta }_1$.

$(\mathit{m},\mathit{n})$	r	(${\mathit{R}}_1,\mathit{\dots },{\mathit{R}}_{\mathit{r}}$)	MLE			MCMC
			MSE	Length	CP	SE	LINEX		Length	CP
							a = −3	a = 3
(10, 10)	5	$(0_{\left(4\right)},15)$	0.9612	26.2292	0.852	0.9550	0.9551	0.9520	0.2291	0.892
		$(0_{\left(3\right)},2,13)$	1.2547	25.5846	0.842	1.1149	1.1159	1.113	0.2237	0.881
	10	$(0_{\left(9\right)},10)$	0.5816	23.2611	0.861	0.5714	0.5722	0.5715	0.1908	0.902
		$(10,0_{\left(9\right)})$	0.4607	17.251	0.852	0.4558	0.4566	0.4550	0.1456	0.914
	15	$(0_{\left(14\right)},5)$	0.4317	18.9238	0.841	0.4224	0.426	0.4211	0.1534	0.891
		$(5,0_{\left(14\right)})$	0.3999	15.2534	0.854	0.3961	0.3963	0.3951	0.1101	0.902
(10, 20)	10	$(0_{\left(9\right)},20)$	0.1865	22.0342	0.844	0.1823	0.1848	0.1822	0.1771	0.892
		$(20,0_{\left(9\right)})$	0.6976	16.9006	0.841	0.6963	0.6967	0.6960	0.1396	0.882
	15	$(0_{\left(14\right)},15)$	0.1499	22.0617	0.852	0.1455	0.1490	0.1450	0.1413	0.901
		$(15,0_{\left(14\right)})$	0.2790	21.6088	0.862	0.2656	0.2663	0.2639	0.1562	0.902
	20	$(0_{\left(19\right)},10)$	0.2695	19.9988	0.847	0.2650	0.2660	0.2640	0.1406	0.885
		$(10,0_{\left(19\right)})$	0.4284	17.5669	0.833	0.4270	0.4282	0.4261	0.1489	0.885
(20, 10)	10	$(0_{\left(9\right)},20)$	0.1083	20.3239	0.852	0.1074	0.1075	0.1072	0.1799	0.903
		$(0_{\left(8\right)},2,18)$	0.0896	21.7656	0.865	0.0836	0.0857	0.0825	0.1836	0.911
	15	$(0_{\left(14\right)},15)$	0.0677	19.876	0.873	0.0665	0.0672	0.0661	0.1531	0.922
		$(15,0_{\left(14\right)})$	0.3867	14.6703	0.843	0.3738	0.3798	0.3718	0.1334	0.893
	20	$(0_{\left(19\right)},10)$	0.2405	15.4809	0.856	0.2364	0.2384	0.2354	0.1453	0.915
		$(10,0_{\left(19\right)})$	0.1099	16.3582	0.862	0.1063	0.1078	0.1058	0.1329	0.902
(20, 30)	20	$(0_{\left(19\right)},30)$	0.2229	18.5122	0.865	0.2174	0.2183	0.2166	0.1814	0.904
		$(30,0_{\left(19\right)})$	0.3389	14.4719	0.856	0.3371	0.3382	0.3365	0.1188	0.895
	30	$(0_{\left(29\right)},20)$	0.0882	16.7545	0.867	0.0602	0.0797	0.0506	0.1371	0.916
		$(20,0_{\left(29\right)})$	0.3981	12.4497	0.842	0.3805	0.3855	0.3803	0.1074	0.896
	40	$(0_{\left(39\right)},10)$	0.3263	13.1275	0.852	0.3048	0.3061	0.3032	0.1185	0.895
		$(10,0_{\left(39\right)})$	0.3646	12.9563	0.847	0.3546	0.3551	0.3541	0.0979	0.887
(30, 20)	20	$(0_{\left(19\right)},30)$	0.0768	14.7231	0.864	0.0731	0.0746	0.0730	0.1286	0.936
		$(30,0_{\left(19\right)})$	0.1656	15.4289	0.854	0.1635	0.1646	0.1623	0.1244	0.902
	30	$(0_{\left(29\right)},20)$	0.0857	14.0271	0.866	0.0844	0.0855	0.0841	0.105	0.923
		$(20,0_{\left(29\right)})$	0.0419	14.5257	0.875	0.0415	0.0416	0.0414	0.1279	0.932
	40	$(0_{\left(39\right)},10)$	0.1187	11.7094	0.852	0.1137	0.1154	0.1121	0.0987	0.901
		$(10,0_{\left(39\right)})$	0.1589	12.2456	0.842	0.1566	0.1569	0.1556	0.1048	0.891
(40, 50)	40	$(0_{\left(39\right)},50)$	0.0659	15.594	0.862	0.0632	0.0637	0.0630	0.1075	0.904
		$(50,0_{\left(39\right)})$	0.0887	14.9784	0.853	0.0883	0.0885	0.0871	0.1333	0.912
	50	$(0_{\left(49\right)},40)$	0.0561	12.6804	0.875	0.0531	0.0534	0.0530	0.0917	0.936
		$(40,0_{\left(49\right)})$	0.1851	13.3252	0.862	0.1840	0.1845	0.1834	0.1146	0.923
	60	$(0_{\left(59\right)},30)$	0.0653	10.738	0.877	0.0641	0.0642	0.0640	0.0974	0.932
		$(30,0_{\left(59\right)})$	0.1029	11.1368	0.855	0.1023	0.1027	0.1021	0.1136	0.894
(50, 40)	40	$(0_{\left(39\right)},50)$	0.0295	14.4829	0.852	0.0284	0.0288	0.0282	0.1235	0.903
		$(50,0_{\left(39\right)})$	0.0489	12.5382	0.852	0.0474	0.0477	0.0472	0.1015	0.901
	50	$(0_{\left(49\right)},40)$	0.1476	11.3295	0.845	0.1455	0.1459	0.1451	0.1104	0.914
		$(40,0_{\left(49\right)})$	0.0925	10.5042	0.865	0.0911	0.0919	0.0903	0.0896	0.934
	60	$(0_{\left(59\right)},30)$	0.0108	11.052	0.892	0.0103	0.0105	0.0101	0.0909	0.941
		$(30,0_{\left(59\right)})$	0.1046	11.0176	0.854	0.1011	0.1017	0.1010	0.0881	0.903
(50, 60)	60	$(0_{\left(59\right)},50)$	0.0598	12.0486	0.862	0.0593	0.0595	0.0592	0.0938	0.923
		$(50,0_{\left(59\right)})$	0.1797	11.6703	0.852	0.1785	0.1789	0.1782	0.0958	0.904
	70	$(0_{\left(69\right)},40)$	0.0259	12.3316	0.864	0.0243	0.0245	0.0241	0.1009	0.925
		$(40,0_{\left(69\right)})$	0.2238	9.8708	0.845	0.2165	0.2167	0.2163	0.0862	0.916
	80	$(0_{\left(79\right)},30)$	0.0251	11.559	0.863	0.0245	0.0248	0.0242	0.1039	0.935
		$(30,0_{\left(79\right)})$	0.0588	11.2537	0.856	0.0574	0.0576	0.0570	0.0773	0.927
(60, 50)	60	$(0_{\left(59\right)},50)$	0.0137	11.7243	0.852	0.0134	0.0136	0.0131	0.0864	0.903
		$(50,0_{\left(59\right)})$	0.1051	10.7115	0.875	0.1020	0.1030	0.1015	0.0843	0.914
	70	$(0_{\left(69\right)},40)$	0.1182	10.3016	0.865	0.1175	0.1178	0.1171	0.0787	0.921
		$(40,0_{\left(69\right)})$	0.2642	9.1534	0.852	0.2626	0.2629	0.2622	0.0861	0.907
	80	$(0_{\left(79\right)},30)$	0.2254	8.2928	0.844	0.2242	0.2248	0.2240	0.0697	0.897
		$(30,0_{\left(79\right)})$	0.0051	9.8200	0.873	0.0045	0.0049	0.0042	0.0705	0.925

,

Table 10. MSEs, lengths and CPs for parameter ${\theta }_2$.

$(\mathit{m},\mathit{n})$	r	(${\mathit{R}}_1,\mathit{\dots },{\mathit{R}}_{\mathit{r}}$)	MLE			MCMC
			MSE	Length	CP	SE	LINEX		Length	CP
							a = −3	a = 3
(10, 10)	5	$(0_{\left(4\right)},15)$	0.0491	17.8513	0.881	0.0483	0.0488	0.0480	0.1392	0.902
		$(0_{\left(3\right)},2,13)$	0.0086	19.0935	0.873	0.0076	0.0078	0.0073	0.1565	0.925
	10	$(0_{\left(9\right)},10)$	0.0478	20.4166	0.863	0.0473	0.0476	0.0471	0.1799	0.912
		$(10,0_{\left(9\right)})$	0.0177	19.8537	0.884	0.0116	0.0118	0.0114	0.1697	0.922
	15	$(0_{\left(14\right)},5)$	0.0006	18.5268	0.891	0.0004	0.0005	0.0003	0.1547	0.953
		$(5,0_{\left(14\right)})$	0.0543	17.1332	0.872	0.0532	0.0536	0.0530	0.1416	0.921
(10, 20)	10	$(0_{\left(9\right)},20)$	0.0398	24.128	0.852	0.0364	0.0369	0.0360	0.1761	0.902
		$(20,0_{\left(9\right)})$	0.0186	14.8359	0.861	0.0175	0.0177	0.0172	0.1108	0.913
	15	$(0_{\left(14\right)},15)$	0.0013	17.3206	0.894	0.0011	0.0012	0.0010	0.1535	0.953
		$(15,0_{\left(14\right)})$	0.0218	11.7284	0.862	0.0214	0.0216	0.0211	0.1066	0.931
	20	$(0_{\left(19\right)},10)$	0.0016	14.5933	0.891	0.0012	0.0013	0.0011	0.1312	0.942
		$(10,0_{\left(19\right)})$	0.0014	13.5589	0.881	0.0012	0.0013	0.0011	0.0938	0.952
(20, 10)	10	$(0_{\left(9\right)},20)$	0.0080	26.1079	0.872	0.0074	0.0076	0.0072	0.2098	0.921
		$(0_{\left(8\right)},2,18)$	0.0576	24.167	0.862	0.0564	0.0567	0.0561	0.1777	0.931
	15	$(0_{\left(14\right)},15)$	0.0965	20.0585	0.854	0.0915	0.0921	0.0912	0.1376	0.921
		$(15,0_{\left(14\right)})$	0.0111	17.7374	0.872	0.0076	0.0079	0.0074	0.1577	0.945
	20	$(0_{\left(19\right)},10)$	0.0005	20.0369	0.895	0.0003	0.0004	0.0001	0.1647	0.952
		$(10,0_{\left(19\right)})$	0.0346	20.2886	0.878	0.0334	0.0339	0.0329	0.1633	0.935
(20, 30)	20	$(0_{\left(19\right)},30)$	0.0061	17.2173	0.864	0.0059	0.0060	0.0058	0.134	0.894
		$(30,0_{\left(19\right)})$	0.0065	12.5227	0.891	0.0052	0.0055	0.0051	0.1049	0.944
	30	$(0_{\left(29\right)},20)$	0.0875	13.6018	0.852	0.0857	0.0861	0.0853	0.1084	0.902
		$(20,0_{\left(29\right)})$	0.0140	11.8181	0.865	0.0137	0.0138	0.0136	0.0957	0.912
	40	$(0_{\left(39\right)},10)$	0.0137	9.5753	0.864	0.0131	0.0134	0.0130	0.0829	0.903
		$(10,0_{\left(39\right)})$	0.0039	10.0931	0.874	0.0034	0.0036	0.0033	0.0803	0.953
(30, 20)	20	$(0_{\left(19\right)},30)$	0.0198	21.4463	0.882	0.0136	0.0141	0.0131	0.2008	0.921
		$(30,0_{\left(19\right)})$	0.0557	16.1941	0.875	0.0549	0.0554	0.0541	0.1789	0.932
	30	$(0_{\left(29\right)},20)$	0.0081	13.4873	0.902	0.0075	0.0076	0.0073	0.0969	0.942
		$(20,0_{\left(29\right)})$	0.0271	15.1902	0.881	0.0234	0.0237	0.0232	0.1322	0.934
	40	$(0_{\left(39\right)},10)$	0.0733	12.0098	0.892	0.0704	0.0706	0.0703	0.0846	0.942
		$(10,0_{\left(39\right)})$	0.0035	11.0788	0.881	0.0033	0.0034	0.0031	0.1017	0.943
(40, 50)	40	$(0_{\left(39\right)},50)$	0.0501	13.3239	0.872	0.0444	0.0447	0.0442	0.1206	0.922
		$(50,0_{\left(39\right)})$	0.0032	9.1975	0.882	0.0028	0.0029	0.00228	0.0706	0.942
	50	$(0_{\left(49\right)},40)$	0.0007	10.2737	0.894	0.0003	0.0005	0.0002	0.0881	0.953
		$(40,0_{\left(49\right)})$	0.0008	9.1146	0.902	0.0005	0.0006	0.0003	0.074	0.951
	60	$(0_{\left(59\right)},30)$	0.0417	10.0117	0.862	0.0402	0.0403	0.0400	0.0813	0.903
		$(30,0_{\left(59\right)})$	0.0083	8.3529	0.881	0.0072	0.0073	0.0071	0.0621	0.924
(50, 40)	40	$(0_{\left(39\right)},50)$	0.0690	14.0396	0.874	0.0665	0.0669	0.0662	0.1166	0.914
		$(50,0_{\left(39\right)})$	0.0019	12.9736	0.885	0.0013	0.0016	0.0011	0.0976	0.903
	50	$(0_{\left(49\right)},40)$	0.0018	12.1394	0.872	0.0014	0.0016	0.0012	0.1218	0.933
		$(40,0_{\left(49\right)})$	0.0005	10.2437	0.892	0.0003	0.0004	0.0002	0.0833	0.941
	60	$(0_{\left(59\right)},30)$	0.0074	9.7853	0.885	0.0073	0.0077	0.0071	0.0700	0.923
		$(30,0_{\left(59\right)})$	0.0006	9.2874	0.901	0.0004	0.0005	0.0002	0.0772	0.952
(50, 60)	60	$(0_{\left(59\right)},50)$	0.0026	9.1799	0.871	0.0021	0.0022	0.0020	0.0802	0.913
		$(50,0_{\left(59\right)})$	0.0867	9.4657	0.862	0.0834	0.0835	0.0833	0.0659	0.902
	70	$(0_{\left(69\right)},40)$	0.0018	8.5242	0.872	0.0013	0.0015	0.0012	0.0641	0.921
		$(40,0_{\left(69\right)})$	0.0298	7.7204	0.853	0.0293	0.0295	0.0291	0.0643	0.902
	80	$(0_{\left(79\right)},30)$	0.0077	7.681	0.871	0.0071	0.0073	0.0070	0.0651	0.934
		$(30,0_{\left(79\right)})$	0.0574	6.3579	0.852	0.0542	0.0547	0.0541	0.0492	0.914
(60, 50)	60	$(0_{\left(59\right)},50)$	0.0678	10.4351	0.862	0.0672	0.0673	0.0671	0.0788	0.911
		$(50,0_{\left(59\right)})$	0.0076	8.9137	0.882	0.0073	0.0074	0.0071	0.0967	0.935
	70	$(0_{\left(69\right)},40)$	0.0066	9.3827	0.865	0.0054	0.0055	0.0053	0.0849	0.923
		$(40,0_{\left(69\right)})$	0.0065	10.0172	0.892	0.0061	0.0063	0.0060	0.0791	0.946
	80	$(0_{\left(79\right)},30)$	0.0239	8.6086	0.882	0.0214	0.0216	0.0213	0.0712	0.934
		$(30,0_{\left(79\right)})$	0.0007	8.7587	0.902	0.0005	0.0006	0.0003	0.0692	0.951

,

Table 11. MSEs, lengths and CPs for parameter ${\gamma }_1$.

$(\mathit{m},\mathit{n})$	r	(${\mathit{R}}_1,\mathit{\dots },{\mathit{R}}_{\mathit{r}}$)	MLE			MCMC
			MSE	Length	CP	SE	LINEX		Length	CP
							a = −3	a = 3
(10, 10)	5	$(0_{\left(4\right)},15)$	0.0021	1.5943	0.892	0.0014	0.0018	0.0011	0.3126	0.925
		$(0_{\left(3\right)},2,13)$	0.0013	2.3479	0.903	0.0003	0.0009	0.0002	0.3369	0.934
	10	$(0_{\left(9\right)},10)$	0.0037	6.097	0.902	0.0003	0.0007	0.0001	0.5097	0.955
		$(10,0_{\left(9\right)})$	0.0082	2.6736	0.881	0.0011	0.0021	0.0005	0.5757	0.942
	15	$(0_{\left(14\right)},5)$	0.0106	4.2639	0.872	0.0027	0.0040	0.0018	0.5439	0.931
		$(5,0_{\left(14\right)})$	0.0060	1.994	0.884	0.0016	0.0025	0.0010	0.514	0.924
(10, 20)	10	$(0_{\left(9\right)},20)$	0.0105	7.9156	0.864	0.0004	0.0013	0.0001	0.6291	0.945
		$(20,0_{\left(9\right)})$	0.0056	2.6465	0.884	0.0007	0.0015	0.0003	0.5824	0.954
	15	$(0_{\left(14\right)},15)$	0.0035	4.4388	0.872	0.0006	0.0011	0.0003	0.487	0.932
		$(15,0_{\left(14\right)})$	0.0012	1.9358	0.886	0.0002	0.0004	0.0001	0.4497	0.954
	20	$(0_{\left(19\right)},10)$	0.005	4.0161	0.872	0.0008	0.0015	0.0004	0.5236	0.941
		$(10,0_{\left(19\right)})$	0.0023	1.7601	0.881	0.0005	0.0009	0.0002	0.456	0.953
(20, 10)	10	$(0_{\left(9\right)},20)$	0.0109	4.7117	0.872	0.0021	0.0033	0.0012	0.5666	0.903
		$(0_{\left(8\right)},2,18)$	0.0096	5.1465	0.881	0.0015	0.0027	0.0008	0.5844	0.932
	15	$(0_{\left(14\right)},15)$	0.0026	2.426	0.882	0.0008	0.0013	0.0005	0.4373	0.942
		$(15,0_{\left(14\right)})$	0.0039	1.7956	0.875	0.0013	0.0019	0.0009	0.454	0.934
	20	$(0_{\left(19\right)},10)$	0.0082	3.2249	0.862	0.0038	0.0049	0.003	0.4702	0.921
		$(10,0_{\left(19\right)})$	0.0012	1.199	0.875	0.0005	0.0007	0.0003	0.3377	0.956
(20, 30)	20	$(0_{\left(19\right)},30)$	0.0011	3.8056	0.901	0.0001	0.0002	0.0001	0.3692	0.951
		$(30,0_{\left(19\right)})$	0.0038	1.6295	0.897	0.0012	0.0018	0.0008	0.4629	0.937
	30	$(0_{\left(29\right)},20)$	0.0010	2.1066	0.891	0.0002	0.0004	0.0001	0.3293	0.958
		$(20,0_{\left(29\right)})$	0.0053	1.6784	0.861	0.0022	0.0029	0.0016	0.4451	0.927
	40	$(0_{\left(39\right)},10)$	0.005	1.9238	0.872	0.0029	0.0035	0.0023	0.4071	0.934
		$(10,0_{\left(39\right)})$	0.0013	1.0891	0.886	0.0007	0.0009	0.0005	0.3259	0.944
(30, 20)	20	$(0_{\left(19\right)},30)$	0.0067	3.3068	0.881	0.0032	0.0041	0.0024	0.4614	0.942
		$(30,0_{\left(19\right)})$	0.0011	1.3617	0.872	0.0004	0.0006	0.0002	0.3509	0.956
	30	$(0_{\left(29\right)},20)$	0.0039	1.9602	0.862	0.0021	0.0026	0.0017	0.3707	0.931
		$(20,0_{\left(29\right)})$	0.0006	0.8729	0.905	0.0003	0.0004	0.0002	0.2791	0.954
	40	$(0_{\left(39\right)},10)$	0.0064	1.769	0.882	0.0042	0.0048	0.0037	0.3562	0.934
		$(10,0_{\left(39\right)})$	0.0035	1.1359	0.871	0.0023	0.0026	0.002	0.3106	0.912
(40, 50)	40	$(0_{\left(39\right)},50)$	0.0021	3.3935	0.891	0.0009	0.0012	0.0007	0.317	0.942
		$(50,0_{\left(39\right)})$	0.0008	0.972	0.902	0.0004	0.0005	0.0003	0.2925	0.953
	50	$(0_{\left(49\right)},40)$	0.0064	2.6733	0.881	0.0037	0.0043	0.0032	0.3493	0.942
		$(40,0_{\left(49\right)})$	0.0014	1.0747	0.873	0.0008	0.001	0.0006	0.2799	0.952
	60	$(0_{\left(59\right)},30)$	0.0086	2.0037	0.864	0.0055	0.0061	0.0049	0.3464	0.932
		$(30,0_{\left(59\right)})$	0.002	1.0111	0.853	0.0014	0.0017	0.0013	0.2793	0.904
(50, 40)	40	$(0_{\left(39\right)},50)$	0.0007	2.0891	0.906	0.0003	0.0004	0.0002	0.2558	0.952
		$(50,0_{\left(39\right)})$	0.0007	0.8264	0.872	0.0004	0.0005	0.0003	0.2551	0.914
	50	$(0_{\left(49\right)},40)$	0.0018	1.7876	0.882	0.0011	0.0013	0.0010	0.248	0.935
		$(40,0_{\left(49\right)})$	0.0012	0.8407	0.887	0.0009	0.001	0.0007	0.2568	0.948
	60	$(0_{\left(59\right)},30)$	0.0008	1.0412	0.905	0.0006	0.0007	0.0005	0.2245	0.956
		$(30,0_{\left(59\right)})$	0.0001	0.5752	0.885	0.0001	0.0001	0.0001	0.1752	0.947
(50, 60)	60	$(0_{\left(59\right)},50)$	0.0036	2.0351	0.874	0.0022	0.0025	0.002	0.2883	0.902
		$(50,0_{\left(59\right)})$	0.0033	1.0689	0.884	0.0022	0.0025	0.0019	0.2791	0.915
	70	$(0_{\left(69\right)},40)$	0.0022	1.4459	0.874	0.0016	0.0018	0.0014	0.2698	0.903
		$(40,0_{\left(69\right)})$	0.0018	0.8717	0.865	0.0014	0.0016	0.0012	0.256	0.912
	80	$(0_{\left(79\right)},30)$	0.0019	1.0359	0.872	0.0015	0.0016	0.0013	0.2463	0.922
		$(30,0_{\left(79\right)})$	0.0008	0.7122	0.901	0.0006	0.0006	0.0005	0.2096	0.957
(60, 50)	60	$(0_{\left(59\right)},50)$	0.0003	1.0657	0.891	0.0001	0.0002	0.0001	0.1978	0.952
		$(50,0_{\left(59\right)})$	0.0003	0.6822	0.881	0.0002	0.0002	0.0001	0.2022	0.955
	70	$(0_{\left(69\right)},40)$	0.0006	1.0579	0.902	0.0004	0.0005	0.0004	0.2039	0.941
		$(40,0_{\left(69\right)})$	0.0008	0.6494	0.897	0.0006	0.0007	0.0005	0.2085	0.943
	80	$(0_{\left(79\right)},30)$	0.0018	1.0157	0.871	0.0013	0.0014	0.0012	0.211	0.932
		$(30,0_{\left(79\right)})$	0.0003	0.4982	0.892	0.0002	0.0002	0.0001	0.1742	0.951

and

Table 12. MSEs, lengths and CPs for parameter ${\gamma }_2$.

$(\mathit{m},\mathit{n})$	r	(${\mathit{R}}_1,\mathit{\dots },{\mathit{R}}_{\mathit{r}}$)	MLE			MCMC
			MSE	Length	CP	SE	LINEX		Length	CP
							a = −3	a = 3
(10, 10)	5	$(0_{\left(4\right)},15)$	0.0016	6.2083	0.894	0.0013	0.0014	0.0012	0.6158	0.915
		$(0_{\left(3\right)},2,13)$	0.0096	9.8337	0.883	0.0025	0.0032	0.0021	0.6859	0.935
	10	$(0_{\left(9\right)},10)$	0.0008	4.6908	0.893	0.0005	0.0006	0.0004	0.5394	0.942
		$(10,0_{\left(9\right)})$	0.0007	2.1234	0.902	0.0004	0.00006	0.0002	0.5293	0.954
	15	$(0_{\left(14\right)},5)$	0.0006	1.8437	0.882	0.0003	0.0005	0.0001	0.423	0.932
		$(5,0_{\left(14\right)})$	0.0010	2.5511	0.892	0.0002	0.0003	0.0001	0.5224	0.951
(10, 20)	10	$(0_{\left(9\right)},20)$	0.0062	2.1036	0.881	0.0052	0.0055	0.0050	0.272	0.922
		$(20,0_{\left(9\right)})$	0.0014	2.5613	0.872	0.0002	0.0008	0.0001	0.6005	0.953
	15	$(0_{\left(14\right)},15)$	0.0008	3.9991	0.905	0.0003	0.0006	0.0002	0.459	0.954
		$(15,0_{\left(14\right)})$	0.0009	1.7354	0.911	0.0004	0.0006	0.0001	0.5051	0.956
	20	$(0_{\left(19\right)},10)$	0.0005	2.1831	0.891	0.0003	0.0004	0.0001	0.4195	0.942
		$(10,0_{\left(19\right)})$	0.0015	2.2219	0.885	0.0003	0.0004	0.0002	0.4602	0.936
(20, 10)	10	$(0_{\left(9\right)},20)$	0.0117	0.557	0.884	0.0104	0.0109	0.0103	0.1493	0.932
		$(0_{\left(8\right)},2,18)$	0.0068	3.4328	0.892	0.0062	0.0066	0.0060	0.2809	0.933
	15	$(0_{\left(14\right)},15)$	0.0025	3.8983	0.896	0.0022	0.0024	0.0020	0.407	0.942
		$(15,0_{\left(14\right)})$	0.0055	3.5499	0.873	0.0002	0.0003	0.0001	0.649	0.952
	20	$(0_{\left(19\right)},10)$	0.0027	1.7917	0.873	0.0021	0.0024	0.0020	0.3384	0.904
		$(10,0_{\left(19\right)})$	0.0009	2.1324	0.881	0.0005	0.0007	0.0003	0.4893	0.913
(20, 30)	20	$(0_{\left(19\right)},30)$	0.0006	3.9579	0.902	0.0002	0.0005	0.0001	0.3994	0.951
		$(30,0_{\left(19\right)})$	0.0007	1.2261	0.894	0.0004	0.0005	0.0003	0.3916	0.942
	30	$(0_{\left(29\right)},20)$	0.0005	1.9524	0.885	0.0002	0.0004	0.0001	0.3545	0.952
		$(20,0_{\left(29\right)})$	0.0006	1.3091	0.894	0.0003	0.0004	0.0001	0.3687	0.946
	40	$(0_{\left(39\right)},10)$	0.0004	1.3497	0.892	0.0002	0.0003	0.0001	0.3367	0.951
		$(10,0_{\left(39\right)})$	0.0011	1.2468	0.871	0.0003	0.0005	0.0002	0.3574	0.934
(30, 20)	20	$(0_{\left(19\right)},30)$	0.0027	2.6328	0.854	0.0042	0.0047	0.0041	0.2869	0.908
		$(30,0_{\left(19\right)})$	0.0011	1.6773	0.877	0.0003	0.0006	0.0001	0.4266	0.931
	30	$(0_{\left(29\right)},20)$	0.0006	3.164	0.885	0.0002	0.0003	0.0001	0.3958	0.917
		$(20,0_{\left(29\right)})$	0.0004	1.8152	0.872	0.0002	0.0003	0.0001	0.4394	0.935
	40	$(0_{\left(39\right)},10)$	0.0004	2.0809	0.891	0.0002	0.0002	0.0001	0.3868	0.936
		$(10,0_{\left(39\right)})$	0.0009	1.421	0.873	0.0002	0.0003	0.0001	0.4389	0.945
(40, 50)	40	$(0_{\left(39\right)},50)$	0.0009	2.3253	0.828	0.0005	0.0006	0.0004	0.2546	0.952
		$(50,0_{\left(39\right)})$	0.0032	1.6158	0.872	0.0012	0.0016	0.0009	0.3897	0.937
	50	$(0_{\left(49\right)},40)$	0.0012	2.7348	0.862	0.0004	0.0006	0.0003	0.3257	0.943
		$(40,0_{\left(49\right)})$	0.0014	1.2392	0.882	0.0006	0.0008	0.0004	0.3484	0.952
	60	$(0_{\left(59\right)},30)$	0.0009	0.9732	0.903	0.0004	0.0005	0.0002	0.2225	0.951
		$(30,0_{\left(59\right)})$	0.0007	0.8538	0.887	0.0002	0.0005	0.0001	0.2852	0.932
(50, 40)	40	$(0_{\left(39\right)},50)$	0.0005	2.9847	0.876	0.0002	0.0004	0.0001	0.3694	0.954
		$(50,0_{\left(39\right)})$	0.0006	1.2293	0.885	0.0002	0.0005	0.0003	0.343	0.921
	50	$(0_{\left(49\right)},40)$	0.0007	2.5446	0.891	0.0001	0.0002	0.0001	0.3557	0.948
		$(40,0_{\left(49\right)})$	0.0009	1.2113	0.908	0.0002	0.0004	0.0001	0.3743	0.958
	60	$(0_{\left(59\right)},30)$	0.0008	1.616	0.887	0.0002	0.0004	0.0001	0.3395	0.946
		$(30,0_{\left(59\right)})$	0.0003	1.0293	0.893	0.0001	0.0002	0.0001	0.3197	0.942
(50, 60)	60	$(0_{\left(59\right)},50)$	0.0004	1.8563	0.884	0.0002	0.0003	0.0001	0.2882	0.948
		$(50,0_{\left(59\right)})$	0.0009	0.735	0.876	0.0003	0.0005	0.0002	0.2527	0.916
	70	$(0_{\left(69\right)},40)$	0.0007	1.3754	0.875	0.0004	0.0005	0.0001	0.2473	0.921
		$(40,0_{\left(69\right)})$	0.0017	1.1828	0.853	0.0010	0.0012	0.0008	0.3181	0.935
	80	$(0_{\left(79\right)},30)$	0.0008	1.153	0.867	0.0004	0.0006	0.0001	0.2405	0.925
		$(30,0_{\left(79\right)})$	0.0040	1.2502	0.852	0.0025	0.0029	0.0022	0.3172	0.928
(60, 50)	60	$(0_{\left(59\right)},50)$	0.0007	0.9208	0.903	0.001	0.0009	0.0011	0.2267	0.957
		$(50,0_{\left(59\right)})$	0.0010	1.2489	0.875	0.0003	0.0005	0.0002	0.3261	0.937
	70	$(0_{\left(69\right)},40)$	0.0008	1.7108	0.905	0.0003	0.0004	0.0002	0.299	0.946
		$(40,0_{\left(69\right)})$	0.0006	1.2422	0.884	0.0002	0.0004	0.0001	0.315	0.932
	80	$(0_{\left(79\right)},30)$	0.0007	1.2609	0.873	0.0003	0.0004	0.0002	0.2768	0.942
		$(30,0_{\left(79\right)})$	0.0007	0.9413	0.881	0.0004	0.0005	0.0001	0.2725	0.957

. Additionally, we assumed informative gamma priors for ${\alpha }_1$, ${\alpha }_2$, ${\theta }_1$, ${\theta }_2$, ${\gamma }_1$, and ${\gamma }_2$ when computing Bayes estimation under SE and LINEX loss functions with hyperparameters $a_i=0.3$ and $b_i=4.0$, $i=1,2,3,4,5,6$. Based on 1000 simulations, we computed Bayes estimates of ${\alpha }_1$, ${\alpha }_2$, ${\theta }_1$, ${\theta }_2$, ${\gamma }_1$, and ${\gamma }_2$ along with $95\%$ CRIs and corresponding coverage probability (CP) using the MCMC method with 10,000 samples while discarding the first 1000 values as ’burn-in.’ We evaluated the performance of the resulting estimators of ${\alpha }_1$, ${\alpha }_2$, ${\theta }_1$, ${\theta }_2$, ${\gamma }_1$, and ${\gamma }_2$ in terms of MSE computed for $k=1,2,3,4,5,6$ and ${\varrho }_1={\alpha }_1$, ${\varrho }_2={\alpha }_2$, ${\varrho }_3={\theta }_1$, ${\varrho }_4={\theta }_2$, ${\varrho }_5={\gamma }_1$, and ${\varrho }_6={\gamma }_2$, as

$$\mathrm{MSE}\left(\widehat{{\varrho }_k}\right)=\sum _{i=1}^M{\displaystyle \frac{{(\widehat{{\varrho }_k^i}-{\varrho }_k)}^2}{M}}.$$

(61)

This process was repeated 1000 times, and we calculated mean values of MLEs, lengths, and CP. The results are presented in Table 7, Table 8, Table 9, Table 10, Table 11 and Table 12.

8. Conclusions

This study utilized the JP-II-CS to compare life tests of items from different units within a single facility. Point and interval estimates for the TPBXIID were generated using diverse methodologies, including maximum likelihood, Bayesian, and parametric bootstrap techniques. However, it should be noted that obtaining explicit MLEs for unknown parameters is not possible, so we used numerical techniques to compute them. Similarly, Bayes estimators are not available in closed form, so we used the MCMC method to compute them for the SE and LINEX loss functions. We tested these techniques on a real dataset and also conducted a simulation study to compare their performance for different sample sizes.

Based on our findings from Table 4, Table 5 and Table 6, we can conclude that Boot-T is better than Boot-P, Boot-BC and Boot-BCa in terms of having the smallest lengths. It is observed that from Table 7, Table 8, Table 9, Table 10, Table 11 and Table 12, the Bayes estimates under LINEX with $a=3.0$ provide better estimates in the sense of having smaller MSEs. It is clear that from Table 7, Table 8, Table 9, Table 10, Table 11 and Table 12, when $m,n$ and r increase, the MSEs and the lengths decrease Additionally, in Table 7, Table 8, Table 9, Table 10, Table 11 and Table 12, the MSEs and CPs of MLE are smaller than those of MCMC. Finally, we found that the performance of Bayes estimates for the parameters $({\alpha }_1,{\alpha }_2,{\theta }_1,{\theta }_2,{\gamma }_1$ and ${\gamma }_2)$ is better than that of MLEs.

This study demonstrated that Bayesian estimators outperformed MLEs in the context of the JP-II-CS for parameter estimation. The Bayesian approach, employing SE and LINEX loss functions, yielded more accurate and precise estimates for the TPBXIID in life testing and reliability analysis. This improvement was attributed to Bayesian estimation’s ability to incorporate prior knowledge or beliefs about the parameters, enhancing accuracy and precision. Additionally, Bayesian estimation provided a complete posterior distribution of the parameters, offering a comprehensive view of estimation uncertainty and variability. The use of the MCMC method within Bayesian estimation efficiently explored the parameter space, capturing intricate parameter relationships.

In summary, Bayesian estimation stands out by incorporating prior information, enabling a deeper understanding of uncertainty, and efficiently handling small sample sizes and sparse data. When applied to the JP-II-CS with SE and LINEX loss functions, Bayesian estimation outperformed maximum likelihood estimation in terms of accuracy and precision, showcasing its practical advantages in life testing and reliability analysis.